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3-16-2021

Quantum Stabilizer Codes, Lattices, and CFTs

Anatoly Dymarsky University of Kentucky, [email protected]

Alfred D. Shapere University of Kentucky, [email protected]

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Repository Citation Dymarsky, Anatoly and Shapere, Alfred D., "Quantum Stabilizer Codes, Lattices, and CFTs" (2021). Physics and Astronomy Faculty Publications. 674. https://uknowledge.uky.edu/physastron_facpub/674

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Digital Object Identifier (DOI) https://doi.org/10.1007/JHEP03(2021)160

Notes/Citation Information Published in Journal of High Energy Physics, v. 2021, article no. 160.

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This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/674 JHEP03(2021)160 Springer March 16, 2021 February 8, 2021 : : December 14, 2020 , and find many : 12 ≤ n Published Accepted Received = c theory, which is based on the root 8 E Published for SISSA by https://doi.org/10.1007/JHEP03(2021)160 a [email protected] This paper is dedicated to the memory of John Horton Conway , . and Alfred Shapere 3 a,b 2009.01244 The Authors. c Conformal Field Theory, Integrable Field Theories understood as an even self-dual Lorentzian lattice. By analyzing all graphs

There is a rich connection between classical error-correcting codes, Euclidean , 8 nodes we find many pairs and triples of physically distinct isospectral theories. [email protected] E 8 ≤ n Department of Physics and505 Astronomy, University Rose Street, of Lexington, Kentucky, KY,Skolkovo 40506, Institute U.S.A. of ScienceBolshoy and Boulevard Technology, Skolkovo 30, Innovation bld. Center, 1, Moscow,E-mail: 121205, Russia b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: CFTs. We consider the ensemble averagepartition over function, all and code discuss theories, its calculate possible thein corresponding holographic a interpretation. self-contained The manner, paper andexplicit is examples. includes written an extensive pedagogical introduction and many interesting examples. Among themlattice is of a non-chiral with We also construct numerous modularexpected invariant functions of the satisfying CFT all partition the function, basic yet properties which are not partition functions of any known Lorentzian lattices, andCFTs thus and define study Narain theirof CFTs. properties. the underlying T-duality We transformations code, dubstabilizer of reduce the code a to resulting can code code theories be equivalences. CFT,by code at reduced graphs. By the to means We level of a study such graph code equivalences, code. any CFTs with We small can central therefore charge represent code CFTs Abstract: lattices, and chiral conformalcodes, field those theories. of Here the weMore stabilizer show type, specifically, that are related quantum real to error-correcting self-dual Lorentzian lattices stabilizer and non-chiral codes CFTs. can be associated with even self-dual Quantum stabilizer codes, lattices, and CFTs Anatoly Dymarsky JHEP03(2021)160 52 54 56 57 58 60 62 62 67 70 17 49 50 48 73 42 n 28 76 – i – nodes 20 8 3 74 ≤ 72 67 n 20 31 31 3 GF(4) 35 1 11 64 lattices and codes − 8 E = 8 = 9 = 12 = 1 = 2 = 3 = 4 = 5 = 6 = 7 8 7 E E n n n n n n n n n n and 7 E A.1 A.2 6.5 6.6 6.7 6.8 6.9 6.10 6.1 6.2 6.3 6.4 4.1 Narain CFTs 4.2 Code CFTs 2.2 Codes over 3.1 Quantum additive3.2 codes New construction A: Lorentzian lattices 2.1 Binary codes E Classification of graphs on B Golay code and LeechC lattice Any Narain CFT isD a toroidal T-duality compactification as code equivalence 7 Conclusions A 6 Enumeration of self-dual codes with small 4 Codes and CFTs 5 Bounds, averaging over codes, and holography 3 Quantum error-correcting codes Contents 1 Introduction 2 Classical error-correcting codes JHEP03(2021)160 ; we verify 32 ≤ ]; essentially, it is n 7 , 6 ], and has been analyzed 10 – 8 . Our work complements these studies n ]. 5 , 4 . It is worth noting that the problem of finding – 1 – 8 becomes with some nuances a modification of the ≤ c n can be bounded from above as a linear programming b d required for modular invariance. The constraints of modu- ]. More generally, classical self-dual binary linear codes are 2 | ) 2 , τ 1 ( η | ]. This relation is not exclusive — there are other known ways in which 3 The connection between CFTs and quantum codes becomes most explicit at the level In view of the fruitful connections between classical codes, Euclidean lattices, and chiral of sphere packing by introducing a new relation between the modular bootstrap and codes. density of a lattice spherea packing version in of a the given sphere-packingfor number problem quantum of with codes dimensions respect [ to ratherrecently the been than distance recast the measure in appropriate Euclideannumerically, terms leading metric. the to CFT improved The bounds modular at sphere-packing bootstrap finite problem [ has whose decoherence a real self-dual“controlling” quantum the code spectral of gap givenoptimization length problem, can which detect. weexplicitly The solve that number numerically the bound for iscodes codes tight with of for length maximum error-correctingfor a capacity given (which code maximize length) is the closely Hamming related distance to the problem of finding the maximum possible lar invariance of themust partition be function satisfied by reduce the toanalogue enumerator polynomial, a of making the set it CFT of possiblethe to modular simple spectral implement bootstrap algebraic a gap “baby” relations for overproblem that quantum CFTs of codes. of given maximizing Thus, error-correcting the capacity, maximization as of measured by the number of qubits of the partition function,enumerator or, polynomial. in the Analogously case tonomial of the of the classical a underlying case, real code, the self-dualLorentzian at (refined) quantum lattice, the enumerator code which level poly- lifts becomes of toappropriate the the the CFT power code’s Siegel partition of theta function function upon of multiplication the by associated the even self-dual Lorentzian lattices.arise from These toroidal latticesfamily compactifications define of of a Narain strings class CFTs.correspondence with of with In a nonchiral quantized other family CFTs B-flux, of words, that CFTs a real of self-dual subset a particular stabilizer of kind, codes the which are we call in code one-to-one CFTs. CFTs, one may wonder if thereall, is a conformal corresponding field hierarchy based theories onexpect are quantum that codes. fundamentally their quantum After relation in todevelop codes nature, this extends and idea, to so include and it quantum showbetween codes. is that an natural In there important this to is class paper indeed we of a will quantum natural codes, and compelling real correspondence self-dual binary stabilizer codes, and with the Leech latticenaturally associated [ with Euclidean evenbosonic self-dual CFTs lattices, [ which in turnclassical give codes rise are to related chiral to chiral theories [ It has been recognized(CFTs) are for deeply many intertwined. yearsconstruction that of Perhaps codes, the the Leech lattices, bestquently lattice played and from known a central conformal the example role extended field in ofas the Golay theories this discovery the code. of symmetry relation the of The monster is the group, Leech Monster which the lattice appears CFT subse- naturally — a particular orbifold of the chiral CFT associated 1 Introduction JHEP03(2021)160 , . ) 8 τ 16 ( R ≤ Z n ; it corre- ], it has been = 7 n 17 , dimensions, in the 10 . This is the lowest- 7 . , 24 7 R of this sort with central = 8 ) n τ ( = Z c = ¯ c ]. There are also examples of rational 11 – 2 – ]. Furthermore, there are simple examples of 14 – 12 reduces to constructing multivariate polynomials obeying ) ¯ τ ]. This observation begs the question of exactly how informa- τ, ( 16 , Z over moduli space of Narian CFTs can be reinterpreted as a sum 15 average has been previously discussed in [ 24 . ≥ 8 which are sums of characters with positive integer coefficients, but which are not c ) ≤ ¯ τ One of our original motivations for the present work came from quantum gravity. In One striking finding is the multitude of inequivalent codes sharing the same enumer- Code CFTs form a discrete subset of the continuous moduli space of Narain CFTs. Another question on which the connection to codes sheds new light is that of the space c , and correspondingly many isospectral CFTs with 8 τ, , ( 8 = ¯ the bulk gravitational theory isa implemented complete in answer the to CFT.codes this within question While an at we important present, do classied we not of have as claim CFTs. at a to This least same have toyshown identified class model that error-correcting of of CFTs the has holography. recently been In stud- papers by two sets of authors [ R the context of the AdS/CFT correspondence,boundary information of is spacetime, understood to inerror-correcting be a codes stored highly [ at the nonlocaltion and is redundant stored form in strongly the reminiscent dual of CFT, and in particular, how the form of error correction seen in sponds to a pair ofdimensional isospectral example even among self-dual lattices Lorentzian associated latticesis with in the analogous stabilizer to codes, and Milnor’sBut in example many unlike ways of the the EuclideanLorentzian isospectral case, case pair where we of the find even next many self-dual dozens example lattices of occurs in pairs in and even triplets of isospectral lattices in way to classify theand equivalence in classes the of process codes find of many interesting a examples. givenator length. polynomial, We which do impliesinequivalent the this isospectral existence for of code many CFTs. examples of The isospectral first lattices such and example appears for CFTs that are T-dualcode to equivalences, we each are other ableform. correspond to Each reduce to canonical a representative equivalent is generalcodes codes. associated code (i.e. with to T-dual an By an code undirected graph, equivalent CFTs) makingknown map code and as use to equivalent of edge graphs of canonical local related complementation. by a The particular representation graph by graphs transformation, provides a convenient yields many thousands of examplesc of “fake” enumerators, already for small central charge We show that this subset,of and discrete hence groups. the space of Acting codes on itself, this can coset be are described as symmetries a relating coset equivalent codes; code discussed later in the textcodes leads which to do many not such correspond examples,the both to latter chiral case, any and for CFT non-chiral, which at withquestion the small all. CFT of central is charge The finding not in connection such known to all or symmetry may not and exist. positivity constraints Atyet that the which must level are be of not codes, satisfied enumerators the by of enumerator any polynomials, code. Solving a simple linear programming problem of solutions ofZ the modular bootstrappartition constraints, functions of namely any modularcharge known invariant CFTs. functions A familyCFTs of with chiral just two characters [ JHEP03(2021)160 4.1 ) and codes 2.1 gauge symmetry. consisting of zeros n n U(1) × ], and also the observation ]. n 10 7 [ , 6 , and the quantum version of the U(1) /c is again crucial — it introduces is similarly mixed. Subsection ∆ is comparable to numerical bounds 4 4.2 GF(4) are called bits. Each codeword encodes is not new, and can be skipped by a /n b 2 d ∈ C c contains both pedagogical and original ma- 3 – 3 – ) as well as their relation to lattices, MacWilliams identities, 2.2 introduces quantum error-correcting codes of the stabilizer type, 3.1 . Components of codewords is a collection of binary “codewords,” vectors of length n 2 discusses classical codes, both binary codes (subsection C Z 2 (subsection C ⊂ introduces a crucial new ingredient — the relation between quantum codes and GF(4) This paper is organized as follows. It includes an extensive pedagogical introduc- Quantum code CFTs are a small subset of the space of all Narain CFTs, but our results 3.2 A binary code and ones, a particular message. Whencertain sent bits over a may noisy be channel, changedto a to make codeword their may it opposite be values. possible corrupted, The i.e., to encoding restore procedure is the designed original form of the codeword and thus recover the We start by reviewingunderstanding classical quantum error-correcting codes codes, and focusing theirwe relation recommend on to Elkies’s aspects CFTs. comprehensive important yet For for a concise more review in-depth [ 2.1 treatment Binary codes familiar with toroidal compactifications.the basic Subsection elements ofsections our contain construction original material. relating quantum codes to CFTs.2 All subsequent Classical error-correcting codes tion Lorentzian lattices. Astart reader reading with the background paper inintroduces from both this Narain classical subsection. CFTs, and Section andquantum quantum codes, is codes intended can but for no readers prior with exposure a to background String in Theory. classical It or can be skipped by anyone tions, most of thereader material with sufficient presented background. interial. section Section Subsection their relation to self-orthogonal classicalMacWilliams codes identities. over Most of it can be skipped by the knowledgeable reader. Subsec- stripped-down setting for studying holographic phenomena. tion. Section over Hamming and Gilbert-Varshamov bounds, and other related questions. With some excep- indicate that they mightwe be cite representative in the a facton certain that the sense. ratio our As of numericalthat evidence the bound the for spectral average on this gap over claim, Future a to explorations class the may of central uncover further code charge indications CFTs that appears code to CFTs have can a provide holographic a useful, interpretation. is dual to aThis three-dimensional suggests gravitational that theory if we with should are consider looking what for happens aa when holographic class we version of average of codes over our codes.note to code the obtain construction, We possibility the we perform of averaged the partition reinterpreting average the function over partition of function the as corresponding a CFTs, and sum over handlebodies. over three-dimensional topologies. The authors conjecture that the moduli-averaged CFT JHEP03(2021)160 , ] . , ] n d G (2.3) (2.2) (2.1) n, k, d [ denotes n, K, d [ ] zeros. We x [ . There are n defined such } 1 , H 0 { , which counts the logical bits into generator matrix k k k bits, where with the closest proper dimensions. To design between two vertices is that are different. The × is a proper codeword (all , obtained by adding the n n 2] d c 2 / c / ∈ C 0 1) ∈ C c is said to be of type 2 . We use the notation − binary matrix of maximal rank. and , d k , c d F n 1 1 , the Hamming distance between c F c . n 2 × . In other words, a classical linear that encode ) ) Z ∈ 2 is one with the maximum possible . = [( k d ) if and only if t c , c ∈ C ∈ − K 1 2 c n 2 w( ( c ( = 0 d , c consisting of two elements 6=0 and + 1 , x c ,c ∈C n 2 1 n Hc 2 c Z c F ∈C – 4 – c 6= 1 = ∈ c , and F ) = min as the sum of all elements of a code vector (with the C G x ) = min ( ) approaching a finite limit, the maximum possible ratio = 0 C d c ( ) n ) = d w( 2 x is the smallest Hamming distance between any two distinct ( HG c d K/ ( 2 , but the exact limiting value is not known. . log x codewords with Hamming distance , so that d/n ) distinct codewords for some nonnegative integer ≤ . K G 2 k ` is the number of corresponding bits in = 2 ) 2 K ) = Im( , c H 1 ( c is a vector space over the field ( d C controls the amount of information which can be sent over a noisy channel. There are A linear code can equivalently be specified by a “parity check” matrix Linear codes always include the zero vector, i.e. the vector consisting of A code is linear if the sum of any two codewords Codewords can be visualized as vertices of a unit cube in Colloquially, an optimal code for given goes to infinity, with introduce the Hamming weight sum taken using conventionalminimal algebra, Hamming weight not of mod all 2). non-trivial codewords Then the Hamming distance is the physical bits. that Ker algebra is mod 2). The parity check matrix is an number of “logical” bits. All codewords are specified by a binary where matrix multiplication isto performed describe over the linear field codes with Hamming distance distance squared components modulo 2, iscode also a codeword necessarily a good code, onesure should they place are as locatedcalculated many far either points with away at from the thetravel Manhattan each cube’s along norm other. the vertices (the edges as minimum The to distance possible, distance get an making from ant one would vertex need to to the other), or equivalently, the Euclidean i.e. one which cann correct errors involving thed/n maximum possible numbernumerous of bounds bits. on When A code containing Such a code canthe correct greatest an integer error corrupting up to codeword, defined with somethe appropriate norm. Hamming distance. The mostthem Given widely used two norm vectors isHamming distance known of as a code codewords message. This is done by replacing the corrupted codeword JHEP03(2021)160 . (2.8) (2.6) (2.7) (2.4) (2.5) ). For 2] = 1 / 2.4 1) . Its parity − n d = zeros and a one , the error can be 6 d = [( e t + c improves the quality of = d 0 c → ]. A violation of that condition . c 18 ,       . e . 0 0    ) i } times: 1 n He. He ...... n {z . Clearly, these undetectable errors must = = ,..., 0 1 1 0 0 | (1 ∈ C A more interesting example is the Hamming Hc Hc – 5 – ≡ bits or fewer, one can restore the original message e ... 1 ~ 1 0 1 00 1 1 0 1 1 00 0 0 1 0 1 1 1 1 1 2] ) = ) = 0 0 1 10 0 1 1 0 / =    bits. Therefore, increasing c c ...... ( ( 1) T ) y y =       code. C G − ( n The repetition code is perhaps the simplest example of a = H d matrix 3] [( , n H be a one-bit error, a vector consisting of 4 , × 7 [7 , has two codewords and Hamming distance 1) ≤ , and is therefore not very efficient. n i logical bit by repeating it i − /n to the closest integer. This code has a small ratio of logical bits to n has been corrupted by some error ≤ ( = 1 c = 1 1 /n ) k 0 c for k/n and therefore can detect and correct any one-bit error, w( i could mean that an undetectable error has occurred, one for which the error e , i is a proper nonzero codeword, = 3 e = 0 , an error has occurred. However, the converse is not necessarily true. A vanishing -th position. Such an error can be detected and uniquely identified via ( code, defined by the following parity check matrix e d i y + 3] c The name of the parity check matrix comes from its role in detecting errors. Typical If a codeword 6= 0 , 4 y = , 0 It has Indeed, let in the c [7 If an error occurs that corrupts by rounding physical bits, Example: Hamming This code, denoted as check matrix is the by an additional physical bittakes that the has value that no makes effectis the on an sum logical indication of operations, that all which nine a automatically bits hardware even errorExample: [ has occurred. repetition code. code. It encodes vector simultaneously corrupt at least the code by making it less likely for undetectablearchitectures for errors semiconductor to computer memory occur. supplement each byte (8 bits) of memory detected by applying the parity check matrix If result JHEP03(2021)160 , ] i c 2 0 ) , Z Z 2] such k, d (2.9) / (2 (2.11) (2.10) , with / l n 1) − 1 Z C ], known R − d n, n =1 [ t l 20 , . bits or fewer. making them P of all possible Γ 19 . A code satu- = [( y 2] k 1 t uniquely charac- ⊂ / ) ) − 1) C C k and − − Λ( Λ( code is a perfect code; n n d 2 as a lattice in . ⊂ 3] is its lattice dual, and , is the volume of this ball, ) simply states that since ∗ Z = [( 4 centered at the codeword ) Γ , Z t ∈ C} 2.9 t bits, would be annihilated by 2 [7 c is the lattice quotient t, n . √ ( . All codewords of a given code ∀ k 2 n bits and therefore , − V Z as the equivalence classes of even ) = n in terms of l Z t < d 2 } ∗ d 2 2 1 Γ ∈ C} ≤ , , √ 0 2 { )! , c mod 2, l . Then √ t = ( 0 (mod 2) ! = / C − modulo shifts by elements of the sublattice ∗ n Z ≤ 2 n Γ ≡ unambiguously indicates which bit should be Z ) Z !( 0 l : – 6 – c ) , one can define its dual, which is an 0 ] · (mod 2) Γ Γ = c c, c t ( ˜ c =0 ( . balls can not exceed the total volume of the space l c X y ∗ d , k . Γ , linear binary codes are in one to one correspondence ≡ n 2 2 Γ / n, k, d n [ Z ) := Γ is a lattice. It is easy to see that centered at each of the codewords of a given code should ⊂ with , v ∈ ) n 0 Λ t, n C ˜ 2] c errors affecting exactly . Then if c ( Z | / 2 ⊂ ˜ Λ( c V )! ∈ 1) √ l { ∗ / v Γ − 1 | − = d 2 n ⊥ . Otherwise different errors would yield the same √ C !( y = [( /l v/ and therefore can be thought of as an equivalence class of lattice points ! t { n n modulo shifts by vectors in ) . 2 satisfying n = n of even integers being a sublattice. Then ) = Z 2 ( Λ C l 2) n Z is a linear code, 2 ∈ C √ Λ( C / c . In other words, for given Z C For a given linear code of type This identification is the basis for a construction of Leech and Sloane [ It is useful to think of the elements of Let us assume that a code can correct any error affecting Γ = ( . For the sake of mathematical elegance (and for reasons explained below) we will rescale , leading to a contradiction). We therefore find the following bound on Z terizes with lattices code consisting of all codewords orthogonal to in give rise to the following set of points in Provided both of these latticescan by be thought of as the quotient as Construction A, whichvector associates in a lattice to any binary linear code. A codeword is a and odd integers.the We lattice can further viewi.e. the equivalence classes set of of2 lattice all vectors integers in to be the seti.e. of the all total codewords numberthe of balls codewords of it radius contains.not overlap, The the bound total ( of volume codewords of all This is known asrating the the Hamming Hamming bound. bound It isthe constrains called repetition perfect. code The is Hamming We can not. define a The ball Hamming in bound the has space a of simple codewords geometric with interpretation. radius errors overall. This numbernon-trivial should values not of exceedindistinguishable the (and total their number sum,H which would affect written in base 2.flipped to Thus restore the the valuenoted) original of is message. to be All understood of mod the 2. algebra aboveThere (except are where explicitly is simply a binary 3-vector whose components are equal to the digits of the number JHEP03(2021)160 . . 1 ~ n ) is + 2 c 1 , is ) and . In 2 n/ I ⊥ , k 2 2 (2.12) C √ 2 = / = w( n vectors, √ n Λ( k / c + e n 1 · ~ 1 2 1 ~ n [ codewords is , c k 2 introduced above 2 , its scalar product and vice versa. The 2 n √ Z / C 1 belongs to the code, and ∈ 1 ~ c of all of its . Even but not doubly-even We apply Construction A to ) II c . includes the vector 2 n w( Λ , and self-duality implies 2 n/ . . One of these vectors, say ∗ n ≤ ) is a linear span of the following R C k Λ is the parity matrix of – 7 – of a self-orthogonal code is an integral lattice. ) = Λ( ⊥ can be combined together into a new . ⊥ C ) . Rescaling by the factor ] 2 C C 2 C √ / Λ( , d Λ( 2 1 , , k ) . This is the checkerboard lattice, isomorphic to the root 2 C 1 n [ is defined to include doubly-even codes as well, in which case − Λ( is a basis vector in I n ). The corresponding lattice 2 i and e . ≤ ] 2.5 n 1 i ( R is orthogonal to all codewords (with algebra mod 2) and therefore , d n ≤ i 1 ⊂ 1 ~ 1 has the following special property. For any , k Γ , is the original code , where series rescaled by 1 2 1 2 ~ code. A code which is not a composition is called indecomposable. The n n [ ⊥ √ √ C B / )] / i i 2 e e 2 2 , d 1 must be divisible by four. In fact, doubly-even self-dual codes can exist only for . At the level of lattices, d (taken using conventional algebra) is equal to the Hamming weight of ⊥ n and 1 ~ Two codes The vector Binary doubly-even self-dual codes, which correspond to even self-dual lattices, are A binary code is called even if the Hamming weight A binary code is called doubly-even if the Hamming weights of all codewords are divis- A linear code is called self-orthogonal if, as a linear space, it is a subcode of its dual, 2 min( √ vectors , 2 / the repetition code n linearly dependent and can be1 dropped. Thus lattice of the divisible by eight. k Construction A lattice of a decomposable code isExample: a direct repetition sum code of and two checkerboard lattices. lattice. with For any even code, belongs to the dual code.hence If a code is doubly-even and self-dual, said to be ofself-dual type codes, II; which the correspond classsome to of treatments, odd the type lattices, class II areit codes of corresponds is type to denoted I the and set are of in all the integral class unimodular lattices. ible by four. The correspondingboth lattice for is codes then and even. lattices,(lattice It that is is integral), any and doubly-even then vice code an versa.even elementary (any We consequence, even self-dual therefore lattice) arrive codes is at are self-orthogonal theare following in sublattices conclusion: one of doubly- to one correspondence with even self-dual lattices, which even. Since all codewordsare of necessarily a even. self-dual Atany code the lattice are level vector self-orthogonal of is (mod integer. lattices, 2), when self-dual the codes code is even, the norm-squared of C ⊂ C A code is calleddual self-dual (unimodular). if Self-orthogonality it requires Therefore is self-dual equal codes can to exist its only dual; for its even values corresponding of lattice is then self- code dual to is necessary for thedual following (in fundamental the property: lattice the sense) lattice to of the dual code The generator matrix of the dual code ~ JHEP03(2021)160 is B ≤ 2 i i (2.14) (2.15) (2.13) ≤ and last 1 k , are linearly 2 2 √ is an arbitrary / √ ) / i Q is the subgroup of +1 e i 2 ) e . In the special case C binary matrix. The 2 + ), the binary matrix ) √ i ), includes an additional k / e , 1 ( Aut( 2.15 − 2.7 elements, but some of them ! 6= 0 n ), the generator matrix of any n ( ) Q The code dual to the Hamming . × 2.14 k B det( ) which permute the first appearing via Construction A. These (all other vectors . matrix with only one non-zero element 2 ,  2 , code yields the root lattice of Lie alge- ) invariant. 2.14 √ n Z √ lattice. / C code. Its generator matrix is given by the 4] / B) is some 1 × series rescaled by , | n, n 7 3 e 4] n root lattice includes integer and half-integer n , B E , . 2 O( C 3 7 – 8 – [7 , = ( I ∈ is non-degenerate. The matrix E coincide, reflecting that the repetition code [7 A.2 T 2 O G 1 1 1 0 0 0 0 ), and √  / 2 2.6 is a binary ], and has been used to classify all inequivalent codes for code and C , and 1 O 23 are called equivalent if they are related by a permutation of 4] , 0 , and OGQ, O 22 C 3 , 2 ∼ 0 [7 √ and / identity matrix and G 2 binary matrix (all algebra mod 2) without changing the code. Usually C ]. The relation between codes and graphs provides useful way to ana- k ) is not unique; one can still simultaneously permute the rows and B k 21 × × ). Its parity check matrix, besides the rows of ( 2.15 k k 2.7 . The equivalence transformations ( ) are mapped to equivalences of graphs under the operation of edge local B 2.14 is the code is known as the Hamming . The conventional basis of the I , the lattices bits act in a more complicated way, see appendix 7 , coming from the rows of ( 3] 1 E Given a code with generator matrix of the canonical form ( By using an equivalence transformation of the form ( The generator matrix of a code is not unique. It can be multiplied from the right by any The lattice of the dual code includes vectors of the form k , 4 = 2 − − , can be used toalences define ( an unoriented bipartitecomplementation [ graph. At thelyze and level design of new graphs, codes code [ equiv- where representation ( columns of n may act trivially. The automorphismpermutation group group of which a leaves particular the code given code code can be brought to the canonical form where the permutation matrix in each row, whichnondegenerate is binary equal matrix. to The full equivalence group has non-degenerate the particular order ofTherefore the two bits codes — thebits. components At of the the level of codewords generator — matrices does not matter. Construction A applied tobra the Hamming coordinates, which does not matchlattices the are factors isomorphic, meaningthis that isomorphism they explicitly are in equivalent appendix up to a rotation. We establish [7 transpose of ( row (which can be chosen in more than one way), n dependent). This is then root lattice of the self-dual. Example: Hamming JHEP03(2021)160 , ). = I 2.15 (2.19) (2.20) (2.21) (2.16) (2.17) (2.18) T BB ) to the original , 2.14 The extended Ham-       ). Such a code is said 2.21 .  to the canonical form ( 1 1 1 0 0 1 11 1 0 11 1 1 0 1 y . lattice. 2 8 T       − 8 code by extending all rows of its . E √ G ) = x . c E . It is the unique doubly-even self- 3] , y . 8 , w( 2 y 2 e 4 y − 2 ) , √ √ + c ,B x . √ / [7 . x , with positive integer coefficients and w(  ! n − B) I  → ) we obtain | n | ) to bring T C 4.2 I x T B (I code and W B 2.13 ∈C – 9 – 2.14 k  , y ∼ c X 4] − 0 y , 2 2 I = 2       4 + . n/

) = √ , H code is denoted x must generate the same code, and therefore ) and ( [8 = = I x, y 4] H ( → , Λ 2.7 T ) = 2 is square. In this case the parity check matrix is given by codewords. To summarize information about its spectrum C 4 x , k B W 2 x, y BB and [8 ( . Under the operation of code duality, the enumerator polyno- ⊥ G k C is not equal but is equivalent in the sense of ( 1 0 1 00 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 W ⊥       by one bit, assigning it the value such that the Hamming weight of . Via Construction A, it gives rise to the unique even self-dual lattice C code has 1) = 2 T = ] , G , the matrix T = 8 (1 k C G n W n, k, d code is obtained from the Hamming = 2 [ , n 4] , 4 , . A generalization of the relation between codes and graphs to the quantum case is 0) = 1 [8 , is a homogeneous polynomial of degree 24 The extended Hamming A linear When (1 C C ≤ When the dual code code, its enumerator polynomial must also be invariant under ( W mial transforms according to the MacWilliams identity In other words enumerator polynomial of a self-dual code must be invariant under of Hamming weights it is convenient to define the enumerator polynomial W It can be easily checked that dual code with in eight dimensions — the root lattice of the Lie algebra where we used the equivalence condition ( Example: extended Hamming ming generator matrix each row is even. Starting from ( understood mod 2.matrix The of same the conclusion Construction A follows lattice, from the explicit form of the generator an important part of our discussion in section When the code is self-dual n JHEP03(2021)160 . . 2 .) 4 ) 0) n/ )) , 24 4 g y (2.24) (2.23) (2.22) − and the = (1 ,W 8 4 1) = 2 ) and also are them- c . Its dual 2 e , x i 2 ( 8 W e (1 √ ( C 2.21 xy / W P ( 1 W or . ) and and and 8 24 e 8 2 y . i e 2 8 . + y W √ Golay code, introduced W ,W in any conventional way, 2 i + 16 ⊥ 8] y 4 C , W 8 is even. Schematically, y ( . The enumerator polynomial x , 4 12 n 2 P x , . The enumerator polynomial of πiτ . All such polynomials lie in the π/ 2 . (The condition ⊥ [24 e iy C + 759 + 14 = 8 = 12 → x y ). C 0) = 1 y 12 , = , and . The same argument applies to the dual ) x formally self-dual 8 ) 8 (1 2.21 e ⊥ e C , q ⊥ C of an isodual code is isomorphic to its dual, 2 ⊂ W C W / W ) and ) 2 | C – 10 – Λ( v | is a polynomial in + 2576 q generated by 2.21 8 ) Λ( and ⊂ ) = Λ( Λ y ) ) The simplest example of an isodual but not self-dual C 8 ∈ 2 X e 16 v C x, y y iso-dual x ( Λ( which are not equivalent to C Λ( ,W + , i.e. they exist only when it is sometimes convenient to use C ) = 2 ] ⊂ ). Such codes are called formally self-dual. All formally W 2 i τ generated by x ( , d 24 W + 759 code. g Λ 2 ) ( = 2.21 code, respectively. Θ 24 is characterized by its theta-function, 24 W P 2 1] n/ g , is invariant under ( when i x , 4] Λ , = 1 W xy ,W = 4 , and self-dual 8 , e + 8 [2 24 [8 e 2 n, k g W [ 0) = 1 code, which includes one trivial and one non-trivial codeword x ( , W W P 1] = (1 since the code is even. All polynomials invariant under these symmetries , C 1 y , W W is enumerator polynomial of the extended [2 24 → − is invariant under ( g y C W An arbitrary lattice Not all polynomials invariant under appropriate symmetries are enumerator polyno- Any doubly-even formally self-dual code, provided it is a linear code, is automatically Any enumerator polynomial of a self-dual code must be invariant under ( W related to its dual by a rotation. Such lattices are called isodual, in contrast to self-dual In what follows we refer topolynomials. polynomials that satisfy these additional conditions as invariant below. Instead of mials of self-dual codes.integers The and coefficients they of additionally bonafollows must fide from satisfy enumerator polynomials are positive polynomial ring Here extended Hamming self-dual. This follows immediately fromand the fact hence that is an included evencode, lattice in is and its necessarily its integral, dual duala lattice, yielding doubly-even code is invariant under ( This is known asselves Gleason’s the theorem. invariant The enumerator generator polynomials polynomials of the self-dual repetition code lattice coincides with the original latticeof after this rotation code, by under are in the polynomial ring Example: isodual code is the This code isintegral not lattice even with and a therefore rectangular not unit self-dual. cell, with The sides of corresponding length lattice is a non- i.e lattices. Finally, there are codes yet self-dual codes are to be isodual. At the level of lattices, JHEP03(2021)160 , ) Z , (2.28) (2.29) (2.30) (2.31) (2.32) (2.25) (2.26) (2.27) . PSL(2 is also the C the function 8 E + 1 . . τ . We define them . 8 πiτ τ y 2 → e + they change as follows πi/τ . For an even self-dual 2 τ 4 = − y /τ e 4 +1 1 x τ = , q ˜ q 2 → / → − . 2 + 14 . The root lattice τ ) . τ , , 8 2) n )) / ) ) x 2 /τ 2 2 Z q 1 /τ, q q +1 ( = ( ( . 1 ∈ n − 2 2 2 ( 2 8 ( θ θ q e , θ Λ 2 2 ) n/ − Θ 2 → − √ √ , ~a q −∞ ) ) + 2 of a doubly-even self-dual code ∞ ) ( ), confirming the modular properties of the . There is a unique modular form of weight 2 2 X = , and under 3 ) C 4 n/ q q n ) θ ( ( C − 2 ( 3 3 ,W ) Λ( q 2.21 C θ θ root lattice. ( Λ( – 11 – )) iτ 2 ∈ , τ ) := 2 W 8 iτ iτ q q − iθ ~a ( ( can be performed independently in terms of Jacobi E − − . Using the Poisson resummation formula, and for 2 2 + 1 2 and ( ) = q → Z √ √ τ ) into a sum over codewords, and for each codeword √ , θ + 2 τ ) iy ) ) = ( ( ∈ ~c 2 ) 2 τ , θ → q C i q code. Therefore, its theta function is given by ( 2 ) = ) = ( is trivially invariant under → ( 2.24 ∗ / a = 2 2 τ 2 8 Λ( 3 2 ) Λ q q y θ e θ n ~v (˜ (˜ τ Θ ( Θ q 3 2 ( 8 θ θ e Λ : Λ Θ −∞ W ∞ → → Θ X = n ) ) 2 2 is a modular form of weight ) = q q τ ( ( ) ( 2 3 τ ) := 8 is a modular form of weight θ θ ( to emphasize that their algebraic combinations can be expanded as power q E Λ ( ) q 3 Θ τ Θ θ ( changes covariantly under the two generators of the modular group 8 ) E τ . We find that ( Θ q Λ Θ remains invariant while we sum over vectors ) For a lattice obtained via Construction A, the theta function can be evaluated as 2 q ( ∈ C 3 Construction A lattice of the The corresponding codetherefore is doubly-even self-dual (so the lattice is even self-dual), and These transformations match theta function associated with the lattice Example: theta function of the Standard identities for the Jacobiθ theta functions imply that under Conventionally, Jacobi theta functions areas functions understood of as functionsseries of in The sum over each component theta-functions follows. We splitc the sum in ( lattice and therefore simplicity assuming the latticedual is lattice unimodular, in terms one of can express the theta function of the When the lattice is even, which is a holomorphic function of JHEP03(2021)160 (2.35) (2.33) (2.34) . Thus, n . In fact Z ) q ∈ ( naturally lead ~a O n becomes would necessarily 4 1 + ) )) C 4 y , . . . . There are several Λ( , it is desirable to leave roots. There are many 8 − 0) 4 x ( 240 with arbitrary n > is the largest possible length . . The overall coefficient can . Then behave as . Another conventional choice ,..., 4 4 8 2 xy 6 ~a 1 b ( c 2 q E E 8 ± b √ 16 = , , = 2 8 2 4 . a / 2 c 8 and 2 ) = ` 2(0 c n τ = 4 + q ( √ lattice is the optimal sphere packing in lattice has 8 + E 4 , 24 1) 8 E a 8 8 for small η 2 b 0) − E Θ E ( 4 + 16 E 8 −∞ ∞ a ,..., → – 12 – X = 0 , intact because they have sufficient length, but n 4 , , the combination and ) 1 )) ) = q 4 ) ± ( τ ∈ C y τ 2 ( ) := ( 2( 4 θ q c 8 − ( . They satisfy E √ , E 4 ) 4 → 2 and the theta series of the Leech lattice introduced below. θ q x Θ ( are polynomials in ( , and therefore . It so happens that y 4 √ 4 ) 8 , θ E τ xy ) ~c/ ( ( q = 2 4 = ( 3 E 2 , indicating that the θ , c ` ) ) 2 ), but it is not completely conventional in the choice of generators. q q → ( ( ]. The discussion above is intuitive but it has a serious flaw: all lattices 3 can be expressed in terms of Jacobi theta-functions. It is customary to x θ , this is simply a consequence of the statement that all modular forms of O 2.22 24 2 6 4 divisible by = + E E n q − , b . This is of course a well-known result in the theory of modular forms. Since ) 3 4 ) = q 24 for ( E lattice and the Construction A lattice of the Golay code. This formulation is the τ η 2 2 ( 8 θ 8 = E E n/ = and 24 Θ η a 4 There is a close relation between codes and sphere packing. An optimal lattice sphere The analog of Gleason’s theorem for theta functions of even unimodular lattices is the = 1 + 240 E 4 , the Eisenstein series design good lattice packingslattice starting from vectors a of good theremove code short form with vectors of thedifferent form ways (constructions) toparticular achieve physical that relevance. result. Here we focus on a construction of eight dimensions [ obtained via Construction A includeno vectors of matter the how form goodhave vectors a of code length mightof be, the the shortest corresponding vectoran lattice in unsuitable approach eight for dimensions, finding but good this lattice observation sphere renders packings in Construction higher A dimensions. To packing requires a latticemaximum with possible a length. fundamental Codes cell of maximalto of Hamming unit such distance for lattices. volume a and given Thinkingthe a of shortest distance codewords vector from as of points thea on good origin the lattice to unit sphere all cube, packing, codewords. such and codes indeed maximize Via the Construction A this should lead to in 1728 weight of generators is provided by Upon substitution Correspondingly the theta series of any even self-dual lattice can be written as a polynomial following. All theta functionsfor for the even self-dual lattices aredirect polynomials analog in of theta ( functions and be fixed byE noting that both ways introduce 4 JHEP03(2021)160 , . ]. 2) Λ 25 +1 √ ∈ from τ . It is (2.40) (2.36) (2.37) (2.38) (2.39) 2) δ is not. 1 / 2 0) → 1 1 Λ Λ τ . Further- is positive ∪ 0 0 2 they change Λ ,..., / ,..., 0 to be odd, to ) 2 , c /τ / 8 1 1 1 . such that w( ± , Λ = Λ n/ 2 2 δ , still belong to the 2( / n/ ) → − √ (5 belongs to ∈ C ac τ 0 )( , c Λ and always changes covariantly 2 , i , . a “twist” (by a half-lattice , any Construction A lattice 2 ) } } is even as well. √ 1 ~ 0 (1 C is odd, and Λ Λ ( √ C ) 0 Λ , ~c/ 2 C Λ } ∈ ∈ 2) W / ( . Under 1 0 / 2 ) Θ c 1 c Λ Λ n/ ∈ − ↔ w( , v , v 2 . , v b ,..., | . Λ can be represented as the disjoint union − , 0 1 2 δ and consider a vector a 2 / removes the vectors 2 ∈ Λ 2 Λ 1 + / Λ ) = 0 δ n/ δ / , ~ ∪ 1 ) c v i 2 is an integer and therefore . Therefore 0 . Under the modular transformation / bc v w( 3 = 0 mod 2 = 1 mod 2 – 13 – → · ≡ { iτb − v v = Λ + ( δ ( δ a − · · 0 2 ) = 1 2 δ δ √ Λ + , i n/ 2 2 ) 1 , | | (1 → δ C ab v v b { { . The “codeword” vectors = Λ W by 8 2 + ( . Choosing this 0 1 = = requires the code to be doubly-even and ) 1 2 is odd, and therefore that n/ Λ n/ C , can be calculated in full generality. Because of permutation 0 1 is integral, Λ − √ , and 2 is a lattice: the sum of two vectors in Λ Λ Λ( 2 / 2) δ Λ 0 + 1 1 ~ Θ 1 + √ iτa Λ τ = − (2 ≥ = is divisible by four. / δ ~ be an integer. The lattice is closed under addition and therefore it is a lattice, while ) change as follows: √ 2 ) → 1 2 ~ C ` c 2 ( 0 0 δ τ is even, , reflecting that the twist does not affect self-duality; however, modular → = Λ Λ w( ) δ ~ Θ /τ C a, b, c 1 Λ( iτc, c is odd, the lattice is even and self-dual. This procedure can also be applied to − 2 → − δ √ τ → . There are several cases to consider: The theta function of the new lattice, obtained from the Construction A lattice by We will call the above construction of a new lattice Let us start with an even self-dual lattice ) be odd is not necessary and is replaced by a c 2 the functions as under invariance under ensure that and even. We spare the reader the details and simply present the answer, If the code is doubly even, lattice provided the twist with symmetry, the contribution of all vectorsw( associated with a given codeword depends only on The twist canoriginal be ones. used Since to any self-dualincludes construct code the includes new vector the lattices codeword the with lattice; they longer are shortest insteadwhich replaced vectors by have length than the It is easy to checkmore that if odd self-dual lattices, yielding aδ new odd self-dual lattice, in which case the conditionvector), that following the nomenclature adopted in the context of 2d conformal theories [ and define a new lattice via Since the original lattice easy to see that We now shift all vectors in We demand that of two sets JHEP03(2021)160 is < ) . In 0 Leech 24 24 (2.43) (2.44) (2.45) (2.41) (2.42) Θ g g ). The , Λ( 24 y 2.23 , confirming ( invariant un- + 24 = I g 20 . y T is invariant under x, y W 4 8 12 x The extended Golay ) ). e 3 BB ac − ( B.1 , that the theta functions ) 16 − . 8 4 y 4 q b ) 8 ( 12 x ) = O ac ( are negative, all coefficients in bc 4 + c ) − given in ( 3 + ( 4 2 + q + 771 , ) ). B 4 x, y 12 24 ( 12 a bc ) η y ab . 2.35 produces the Leech lattice, with theta 12 + ( x Leech 672 4 ) + ( ) = 4 W 24 ) − 2 g ab and ( 24 3 4 ` + 16773120 g code (up to equivalences). It is denoted 2 Λ( E 24 – 14 – + 2558 = ( Λ( g q code and Leech lattice. are positive integers. is doubly-even. The theta function of 8 = Θ 8 The Construction A lattice of W to y c 8] ) )) . = 8 24 2 , 16 = g 24 )) d + q 2) g x 2 ( , 8 12 24 q 2 2 √ Λ( b ( , η 2 , θ Θ (2 + ) / , θ 2 = 24 [24 8 + 771 1 ~ ) 720 q lattice. a 2 4 ( n = 1 + 196560 q 3 y 8 − = in the sense that the new lattice is isomorphic to the old one. ( θ = 3 3 4 ( 20 E 4 δ θ ~ ) it is specified by the matrix x E 2) ( 3 Leech E confirms that √ , = Θ Leech − 2.15 (2 iy 24 Leech W g / 24 1 ~ x W → W Leech = y Θ = = δ ~ expansion reads code is the unique q Leech Leech ) and Θ 8] W — and its shortest vector has length , -expansion of 2 2.21 To summarize, there is a close relation between codes and their enumerator polynomi- The analog of Gleason’s theorem for modular forms from above guarantees that Applying the twist It is a matter of a few minutes of computer algebra to verify that 12 q , ≤ 2 is exclusive. The answerunique is way: no. inequivalent Enumeratornon-isomorphic codes polynomials lattices may characterize can codes, share be but theinvariant isospectral, polynomials not same i.e. which in have polynomial. are a the notare same enumerator self-dual Accordingly, theta lattices polynomials different not series. of related any to There code. any are code Likewise, also via there Construction A, and so on. To see how this We emphasize that whilethe some coefficients of als, and lattices and their theta functions. A natural question to ask is whether this relation can be expressed asder an ( “enumerator polynomial,” i.e. a polynomial in Its small which indicates that the` Leech lattice (famously) has no roots — vectors of length function explicit form of given by which follows from the explicit form of Example: extended Golay [24 the canonical form ( that the code is self-dual, and to evaluate its enumerator polynomial the twist by As a consistency check oneof can the verify original using and identity new lattices are equal Example: twist of the JHEP03(2021)160 . . .) 2 24 24 and 42 , /Z , but 8 − 8 16 2 e e , = W ⊕ define two 8 r = 8 e Spin(32) n + 16 D and . We conclude that 8 2 4 E E ⊕ 8 the story is simple: there is to ensure all coefficients are E , which is not isomorphic to ]. The situation is even more is not known there are various 8 Speaking colloquially, our paper E 31 = 8 147 – 1 d/n is the root lattice of code, but other lattices are related ⊕ n ≤ 29 8 + 24 16 r E g D ≤ , all invariant polynomials can be written 42 doubly-even self-dual codes, and overall − = 24 9 . Thus for 4 n , where – 15 – 2) / 1 ~ via Construction A. The theta series of that lattice, ]. For + ]. Construction A applied to the latter yields the even , there is still a unique invariant polynomial 8 ]. The isospectral lattices 20 e an integer goes to infinity. Analogous questions can be asked about 16 corresponds to the smallest allowed value of 27 28 r n , D ( = 16 24 26 ]. (We refer the reader interested in a quick summary of these [ g ∪ n 36 with + 16 when 16 – d 4 ].) In particular, Euclidean even self-dual lattices can be used to D 3 )) 32 , 4 k/n = y 4 . In this case there are , 3 + 16 − is the unique self-dual doubly-even code, and there is a unique invariant 4 D 8 . There is also a unique even self-dual lattice in eight dimensions, the root x e = 24 8 ( e , n for fixed , which is related to xy W 8 ( = 8 r E n d/n + 8 3 e One of the central questions of code theory is to understand the maximum possible The relations between codes and lattices can be extended to CFTs and their vertex For . Both codes have the same enumerator polynomial, and thus both lattices have the We thank Xi Yin for a discussion on this point. W , is the unique modular form of weight 1 + 16 4 non-chiral CFTs. value of lattices and sphere packing. While the optimal value of The connection to codes provides aIn new particular, angle to the probe fake variousCFT aspects enumerator partition of functions, polynomials 2d modular chiral mentioned invariant functions theories. some above satisfying positive of give conditions, which rise at are least to notextends “would-be” partition the functions of relations any between theory. codes, lattices, and CFTs to include quantum codes and code-less invariant polynomials as “fake.” operator algebras [ relations to table 1define of chiral [ CFTs, which play a prominent role in string theory and mathematical physics. historically, the Leech lattice isto associated each with other the in a similaras manner [ positive. (The GolayMost of code these polynomials are not enumerator polynomials for any code. We refer to such be found in J.different Conway’s heterotic book string [ theories,nuanced related for by duality [ non-isomorphic even self-dual lattices.a particular There code is besides no those simple nine way obtained to via Construction assign A. each In lattice our to discussion, and D same theta function,these the two unique non-isomorphic modular latticesis form are Milnor’s of isospectral, famous weight example sincewith of eight, their equivalent distinct Laplacian theta compact spectra. series spaces (the coincide. An tori excellent defined This nontechnical by discussion these of lattices) this point can a perfect correspondence betweeninvariant self-dual polynomials. doubly-even codes, For eventhere self-dual are lattices, and two inequivalentan self-dual indecomposable doubly-even code codes, aself-dual decomposable lattice code The Construction A lattice of the former code is For even but not doubly-even self-dual codes the situationpolynomial is even more complex. lattice of E works we discuss the most restrictive case of doubly-even self-dual codes for JHEP03(2021)160 . , n 1 6 − n n/ 2 (2.48) (2.47) (2.46) and ≤ d ) should must be ) readily ]. d 2.9 d/n 2.9 46 , that would d in ( codewords and there are and l n 5 K n/ ] and it is expected , . If the corresponding ≤ 11 . 41 , 0 40 d/n ≈ k < d ∗ ). The idea is to fix ≤ 1 is the minimal distance between 2.9 ], which, remarkably, are related known as the Gilbert-Varshamov , p centered at all saturates the linear programming d the Hamming bound ( 45 for 1 d can be increased. Thus for a linear – 2 . k d/n k − ln(2) K 42 − = 0 d , n = 2 2 . Since 24 =0 ) = + 1) n y ∗ K <

j p ) ) ( centered at a given codeword does not include (2 1 , n 1 – 16 – x, y − 1 ( 2 =0 is the Shannon entropy. This bound is suboptimal. Y j − ) − n/ W d p k d y 1 2 ,H ( ∂ ]. − V 10 – → ∞ 8 ) ln(1 p for which the following inequality is satisfied − d , n ]. They can be thought of as simpler, more restricted versions of ∗ (1 p 2 39 − , , but not in general, the bounds obtained this way are tight: the ) n ≤ p 38 ]. These bounds can be further improved [ d n ln( 39 p – − 37 subject to linear constraints and inequalities. If we additionally require that the ) = 8 e p for which the problem of finding invariant polynomials is feasible is also achievable ( in this way, which for type I and II self-dual codes read d H ,W 2 i There is a conceptually different way to obtain a Gilbert-Varshamov bound, suitable Besides upper bounds, there is a lower bound on d/n W for self-dual codes, which leads to essentially the same results. For even This bound can be improvedgo by only noticing over that even for values. evenSimilar codes Furthermore, improvements the the are sum full possible over space alsowe of for disregarded doubly-even even self-duality, codes. codewords but has a In volume generalization the considerations to above self-dual codes is possible [ and put a bound onany the two number codewords, of the codewords ballany of other radius codeword. Weask consider if balls they of covercode the radius we whole find space. the maximal If they do not, unimodular lattice [ linear programming bounds onto sphere modular packing [ bootstrap bounds [ bound, which is closely related to the Hamming bound ( a polynomial is algorithmicallyon hard. Nevertheless onerespectively can [ establish asymptoticthat bounds additional systematic improvements are possible.codes are The parallel linear programming to bounds linear for programming bounds on the length of the shortest vector of a reduced. For small largest as the Hamming distancebound of are a called code. extremal. Codesmay yield The for invariant linear polynomials, which but programming most bounds of these are are not fake, constructive and reconstructing — a they code from even self-dual codes the space ofin invariant polynomials is the linearhypothetical space enumerator of polynomial all polynomials describesimpose a additional code linear of constrains Hamming distance discrete linear programming problem is infeasible, there is no such code and provides an upper bound, where One can derive stronger bounds using linear programming techniques. For instance, for upper and lower bounds. For self-dual codes JHEP03(2021)160 ) ¯ ω c . To . In w( , ω, ω (2.52) (2.53) (2.49) (2.50) (2.51) 1 11 . , 0 0 → . GF(4) = n ω ≈ ) 2 , ∗ iy invariant and for which the − p − 1 ω = 0 d , ≥ x 0 x = is called additive if ω n + ( . We are specifically d/n would then define the = 2 − n F F ) , a fact which will have . x be even, then ] = 11 iy E . b + 0 ω Γ C ⊆ . + C ≈ x 1+ ¯ n, k, d and ∗ [ , yielding p + ( , is smaller than one, then there x, x } ⊂ , consists of four elements a zeros. The Hamming weight n n Z = ) ) n . A code y y ∈ + 1) . As before, the Hamming distance x → ∞ c k < d 2 , all other relations are automatically − + × − n GF(4) a, b x x 2 ≤ 1 | is not dual to . One can calculate the enumerator poly- = 0 is given by the value of n/ 1 8 ] + ( + ( E + 1) , b ω x = GF(4) . For example j n n 4(2 3 2 ) ) 48 2 Γ , the lattice of the so-called Eisenstein integers. + , F (2 , this number includes type II codes) and d/n y y , for = 0 2 – 17 – 2 πi/ 8 a + 1) k 2 − + − { x √ y 47 1 2 =0 − , / k Y j − x x e is said to be of type × 1 n/ 2 − 27 — the unique field with four elements — because of n k 0 2 = = n/ . x √ d ¯ ) + ( ) + ( ω / = 4 n n by requiring that both 2(2 is divisible by 2 x, ) to the averaged partition function of a certain class of y y E A K n and we first consider the triangular lattice in the complex plane = + + = 3 n n 2.51 2 Γ x = GF(4) is divisible by x x rescaled by E ( ( πi/ , meaning that the sum of two codewords is a codeword. Additive = 0 n F 2 Γ 2 2 2 0 + F e x A n/ n/ . There is a conjugation operation which leaves 2 becomes of order one when and size 2 2 = ω d GF(4) d ω y . With the exception of d ¯ ) = ) = ω F, − n ∈ x = 1 + ↔ x, y x, y ( ( is the number of non-zero elements of x I 2 ω II ∀ n ) we will relate ( ω . In contrast to the binary case, W 5 ) W F E = ∈ ¯ ω Now we are ready to define codes over Asymptotically, the lower bound on (2Γ c / is a vector space over E C codes always include the trivialfor codeword consisting of of a code isHamming the distance minimal Hamming weight of all non-trivial codewords, and a code with This is the root lattice If we define aΓ new lattice consequences later. and exchanges satisfied if we take impose the condition interested in codes over their relevance to quantum codes.subject The Galois to field the following relations chiral CFTs. The valuespectral of gap the in Gilbert-Varshamov bound a random CFT from2.2 that class. Codes over Analogously to binary codes, one can define codes over any field So if the sumis of a the code coefficients with of Hamming distance coefficient of section ( type II self-dual codes,nomial when averaged over all such codes [ type I self-dual codes (if JHEP03(2021)160 k is n = 4 F (2.58) (2.57) (2.54) (2.55) (2.56) is even, K ∈ . Additive ∈ C H x, y 4 c . . It is easy to n 2 H+ R and 4 E ⊂ . Otherwise, if some . These two versions 4 i E y includes points outside H+ II is an integral lattice in Γ i as the pre-image of the 4 ∗ ¯ x ) i n . The code is called self- E H+ 2 . C . 4 P R  (Γ n of all codewords y , which requires that for any = F of an even code is even. Even ∈ C} ⊂ = . ) . F 2 H − ) c . An additive code with ∈ k n ) defined above. The MacWilliams C . All linear codes are additive but n , c x y F C w( ) , consists of all codewords orthogonal Λ( 2 c − ≤ ∈ x, y y ∈ C GF(4) ( ⊥ x k 0 w( ) make up to family 2 c C + 3 with all codewords given by → x (mod 2) , x, y G 2.56 i c  ¯ , x y . Linear self-dual codes under the Euclidean i n C ⊥ ≡ , y x F C W – 18 – for some y k + ∈ b ω k = − ~ i 2 n y + 3 C + ]: i Gx x ¯ = 4 x 49 is not self-orthogonal, and , or explicitly – , ~a i ) = 2 n K → . The dual code n ) = E X ) 47 Z x F [ x Γ E ( = x, y c ∈ ( generator matrix . If the Hamming weight y ∗ ⊥ H+ b (2Γ ~ · ) k one can introduce an enumerator polynomial exactly as in the C 4 / products form the code families known as E x n ~a, × must also be a codeword W ) | H (Γ ) E n and self-dual if , ωc b ω ⊂ ), with the Hamming weight ~ (Γ , and are odd, the codes are referred to as odd, or type I, and are in the ( GF(4) ⊥ H E = + ) . 4 c 0 , 2.19 ~a c 2 Γ { . There is also a Hermitian version, E H+ , w( i 4 4 y C ⊂ C i ) = ⊂ ∈ C x . C i c H 4 P Λ( H+ I 4 , although and Hermitian ) = For codes over The Construction A lattice of an additive self-dual code To define duality on the space of codes, we need to introduce a scalar product. There For each additive code we can define a lattice in A code is called linear if it is a vector space over E . It is not self-dual because ) Γ n 2 , x, y Enumerator polynomials of self-dual codes are invariant under family binary case via ( identity relates the weight enumerators ofsame the form original for and dual additive codes; it takes the R of the code is calledself-dual even. codes are The said Construction toof A be the of lattice weights type II, and belong to the family orthogonal if ( codes self-dual under thesee Hermitian that product ( a third, physically relevant scalar product, In all cases the(with algebra is respect over to a given inner product) to all codewords of This is the analog of Construction A for codesare over several natural choices.( First, the so-calledare Euclidean homogeneous scalar and can product be of used to define duality on the space of linear codes. There is code under the map codeword not vice versa. Forlength binary of codes, a the linearcan two code be notions is specified coincide, always by but an not for other fields. The JHEP03(2021)160 ) C . Λ( 1) (2.61) (2.62) (2.63) (2.64) (2.65) (2.59) (2.60) , or over self-dual } 1] Z = (1 , plane inside 1 ∈ T , C G elements in the [1 ! a, b, is the enumerator n | , ) 3! 2 i . Upon rescaling the changes sign. Hence ] , 3) 3 bω } 1 W . The same operation 50 √ √ φ Z ¯ ω + t/ τ/ ∈ a ( . , , , ) or . ) ) C 2( . 3) 3 3 → πiτ { 2 ω Λ( q q 3) a, b, 2 , while y ( ( τ | τ/ e would be covariant under , and within each plane permute 4 4 ( τ/ ) . n θ θ 1 ( = C 2 6 ) ) + 3 1 φ q q y bω R 2 φ ( ( = (1) ) := Θ 4 4 x 2 generated by [ t + 2 ]. θ θ − is isodual, i.e. it is equal to its dual c ) a = ˜ + 18 2 Θ( 2 2 39 4 − 3) + 3 3) 2 / , q 4 and i ) ) + with the generator matrix 1 plane. Alternatively, one can characterize ) is invariant while y 3 3 2 τ/ τ/ ,W 1 3 2 i W q q ( ( 1 0 / F C x ( ( 0 0 1 + 2( ) , φ φ W 3 3 W { φ φ 0 C θ θ ( , ) after rescaling ) ) , it is sufficient to consider each φ – 19 – ) ( 3 3 planes inside q q P Λ( iτ iτ + 45 ( ( and C C √ √ + 1 3 3 6 − − C ) where 2.58 θ θ W if and only if the code (and corresponding lattice) x t in each Λ( τ y 2 ˜ = Θ( + ) = ) = → n 6 ) = ) = ) = + 1 ) π/ h x -modular lattice [ τ τ τ τ /τ /τ it ( ( ( τ 3 are defined to be equivalent if they are related to each other 1 1 also change covariantly under the modular transformation ) W 0 1 = − C φ φ − − 1 1 ( ( → Λ( 0 1 as a , φ W in an arbitrary order. There are a total of Θ φ φ τ 0 ) = ( C 3 GF(4) φ /t 1 πi/ , the theta function for a self-dual 2 τ − − 3 e ˜ Θ( √ , and multiplication of some “letters” by = ¯ ω = after a rotation by t , ω ↔ ∗ 3 ) is the weight enumerator of the hexacode, introduced below. , 4 ω is easy to recognize as the enumerator polynomial of the simplest πi/ / 6 2 1 , in which case it is a polynomial in /τ h 1 e 3 1 y is invariant under / of a self-dual code W W ) ) individually and sum either over the triangular lattice = To calculate the theta series for If the self-dual code is even, the enumerator polynomial is additionally invariant under Additive codes over ) C C n C → − → − Λ( 2 , ω which reflects that(Λ( the rescaled lattice Λ( These transformations coincide withargument ( to Under modular transformation Θ is even. Theτ functions equivalence group. R the triangular lattice shifted by half-vector by a permutation of“letters” their “letters” (componentsshould of be the applied codewords), to conjugation allthese codewords are of of isomorphisms some the which permute code.1 In terms of the corresponding lattice y Here Here additive code with onlypolynomial of one a non-trivial linear codeword “repetition” code over and therefore are in the polynomial ring JHEP03(2021)160 . ψ (2.66) (2.67) GF(4) for this i , an even E 12 K . distinct physical . Only the very  4 n q . Later in the text  4] ). Since it is a linear , O GF(4) 6 , + 2.60 [6 3 q , analogous to results about . + 4032 GF(4) . 2 ⊂ H    q C ω 1 explains the relation of stabilizer codes ω ω 1 ) in twelve dimensions. This follows from ∈ H 2 is the Coxeter-Todd lattice 3.2 ω ω ω ψ ≤ 12 2 = 1 + 756 The hexacode is the unique linear even self-dual is mostly pedagogical; it introduces quantum R 6 1 – 20 – quantum spins, or qubits. Initially the system is φ < ` it would be denoted as ⊂ 1 0 00 1 1 0 0 0 1 3.1 n 0 )    6 in some unpredictable way. This is quantum error. H+ + 18 h 4 = 0 4 1 ψ φ T Λ( 2 0 G φ → except for the first one are divisible by three. ψ 6 + 45 h 6 0 defined by the following generator matrix φ . Its enumerator polynomial is given by ( W 6 H h . Because of unwanted interactions with the environment the system 4 ) = τ ( ∈ H 12 K ψ Θ code of type 4] Interactions with the environment can be described as linear operations acting on There are many other results concerning codes over The Construction A lattice , 3 , qubits at once. Furthermore, the correction of quantum errors is possible only for certain states that belong to a special code subspace More accurately, one should speak ofsimplicity a we quantum will channel acting assume on the ainteractions system density with always matrix, remains the but for in environment a form pure a state. linear Operators space. describing We can choose a basis changes its quantum state We would like to devisebe a quantum protocol error to correction. returnrestrict to the Clearly quantum errors system this of to can a particularqubits, its not type. one original usually be For state. assumes a done system a consisting That random int of interaction would full with the generality, environment so that affects we at must most to Lorentzian lattices, which is the central ingredient3.1 in our construction. Quantum additive codes Let us consider ain system some consisting state of In this sectionLorentzian we integer lattices. introduce quantumstabilizer Subsection codes stabilizer and codes explainslast their and part relation of establish to thisenumerator their classical section, polynomials, codes where relation is original. we over to discuss Subsection real self-dual stabilizer codes and their refined bounds later in theand text binary after quantum making stabilizer codes. the connection between classical codes3 over Quantum error-correcting codes the theta series binary codes, including a series of linear programming bounds. We will present these As an additive self-dualwe code will from refer to itcode, as all coefficients of lattice with no roots (vectors of length [6 Hexacode and Coxeter-Todd lattice. JHEP03(2021)160 , i i ). e E ) it 2.9 (3.3) (3.4) (3.1) (3.2) . 3.4 i E . A code . Assum- C ) . For both C j e qubits is H t + do not overlap, H\H . From ( 2 k C c H = ]. 0 2 ) = dim( c ) = 2 C 52 , C i tensor products of Pauli physical qubits. For ex- H e i H l 3 E + 1 = 5 c . dim( , affecting up to n n dim( = I j. qubits to be affected). Hence the . 0 1 l = c l 6= 3 . ) = 2 1 )! l E H ∈ C ! ] introduced below is a perfect quantum ) does not have to apply — when two − 2 n must not overlap. The total dimension of c 54 n , i 3.1 dim( C !( + l , it is necessary and sufficient that each H i ≤ 1 i = 0 c t ) E ) leads to the classical Hamming bound ( must always be distinct, which is the classical =0 ) is called non-degenerate. E C – 21 – l X qubits is spanned by = 0 2 ]. The reduction of all possible errors to a handful H l 2.1 t, n c 3.1 j j ( e 51 ) = q E V + . In the classical case that would mean that the errors ∩ ) and t, n i C C (all algebra is mod 2), C ( e 0 1 q H 0 2 H H c c physical qubit, one needs V i is reversible, and therefore logical qubit and to be able to recover the state after any E logical qubits, in which case = C in section ( dim( is annihilated by the parity check matrix, meaning that both 0 1 k H = 1 c 0 2 j = 1 , including the trivial one c ]. A code saturating this bound is called perfect. Often the code e t i k + E 53 6= is called a discretization of quantum errors [ i e i 0 1 c E , be nondegenerate (reversible) and that the images of qubits [ C , and assume they are subject to errors, restricted to ). Indeed, if will describe t H i C , to be correctable, E 3.1 ∈ C j H 2 , e i , c will yield the same error correction protocol, which will fail to undo at least one of e 1 j c e The condition There is one important exception when ( The Knill-Laflamme condition has a classical counterpart: correctable errors of a linear subspace follows that to encode quantum error affecting ample, the 5-qubit protocol oferror-correcting Laflamme code. at el. [ all images can not exceed the dimension of full Hilbert space, yielding This is the quantumfecting Hamming up to bound for codes correcting arbitrary quantum errors af- Each error ing the code is nondegenerate, the images There is a quantum versionof of quantum the errors Hamming which bound,matrices affect which (the exactly is identity as being follows.total excluded, number The as of linear we errors space want all errors. The similarity of thei.e. quantum the case comes possibility from to the represent linearity any of quantum quantum error mechanics, asdistinct a errors linear act combination identically of on are the the same, butfor in which the all quantum correctable case errors the satisfy errors ( could act differently on analog of ( In this case the error and of linear operators classical code must produce differentcode results. Consider two codewordserrors of a binary classical to an arbitrary linearrestricted combination to of the This is the Knill-Laflamme condition [ space, a basis of quantum errors. Crucially, to ensure reversibility of quantum errors due JHEP03(2021)160 k + − k 1 = (3.9) (3.8) (3.5) (3.6) (3.7) , and , and n (3.10) i a acting qubits k ψ z k − σ n 0 − spin up, n ⊗ | l ψ 0 = . } k, 1 . I , − 0 { ≡ n 0 = ) ≤ z i i σ ( “logical” qubits ≤ k 1 , a physical qubits, which are isolated , , and will be left intact, while the ) . A quantum error-correcting code k i n C k , a I } . − ) H i ∈ H i n z i , k n 0 . Such a code can correct for any error σ a | − − ( ]] k ). This is easy to do: one simply needs ψ n {z − into 0 . . . a n ⊗ 3.6 ⊗ · · · ⊗ C l includes all states of the form ⊗ | I | +1 ψ H n, k, d l – 22 – k C is the minimal number of physical qubits which [[ ⊗ ψ a ⊗ · · · ⊗ = H z d = σ +1 ψ k i ⊗ | a ψ ⊗ ) k I } z ) in classical case and using it to reconstruct the original σ 1 (( . . . a − 2.4 qubits. i is denoted 1 {z ⊗ + . a 2] k | d C k / ⊗ · · · ⊗ 2 | H = × 1) k i 2 = I = n − , and then applying the recovery operator i , and i C d ) to the desired form ( logical qubits are implemented as g n = I H k . . . a i 3.7 n, k R g . It can be defined as the subspace invariant under the action of 1 = [( k . . . a a t | 2 in the computational basis, called syndrome measurement, and then applying +1 qubits can be represented in the conventional binary basis ( a k ψ a n is the desired result of unitary evolution produced by the quantum computer, while l ⊗ | l ) = ψ ψ k In our example, the code subspace To illustrate how quantum error-correcting codes work, we consider an oversimplified The quantum Hamming distance is some unknown random state resulting from interactions with the environment. This − , is analogous to evaluating ( a n has dimension on any of the auxiliary spins, Measuring R codeword. corrupted state ( to re-initialize theauxiliary auxiliary qubits system. in the Thisonto computational can up-down be basis, done which by will first project measuring the the total state state of example may appear unrealistic becausethe we environment, have physically the isolated systems the are logicaldoes not qubits not entangled, from matter. and therefore All state measurementsour of performed insistence the in auxiliary on the qubits first includingNevertheless, lab the will it be auxiliary is insensitive qubits to instructive in to our ask considerations a is question: inconsequential. can one devise a protocol to bring the will evolve to where ψ Auxiliary qubits will bequantum initialized computer performs in unitary the evolution state of the first Because the auxiliary lab is imperfect, after some time the state of the last in a lab asauxiliary part qubits of a located perfect in( noiseless a quantum different computer. lab We inspin additionally an down) consider imperfect environment. States of all need to be affectedcharacterized to by map aaffecting state up from to situation in which JHEP03(2021)160 . i is l = of k − i . It 2 n g − g , and 0 U in the (3.15) (3.12) (3.13) (3.14) (3.11) n α, β H i ⊗ | . n l ≤ n ) ψ i 2 . . . a Z . ( i to ensure ≤ physical qubits + i +1 ∈ k k 1 n ± a . a . This can be seen n. | 2 i operators acting on . Thus naively only = k for ≤ − n k  − 2 i , α, β ) 1 ) = 0 n − . i i , identify corresponding n 0 or  i | ≤ n g , β 1 β i 1 ) ± Tr(g α z ( ) = 4 logical qubits we would need σ , = k ( } k 0 mod 2  3 − logical qubits with the possibility , , ≡ 2 . The group generated by the k define the basis , i C i t, n 1 = I vectors β ( g , H i · q ⊗ · · · ⊗ 0 g k V j 1 with eigenvalues i † β ) k α + i ) i − ∈ { k − g z =0 a i i − n t (g n σ g j ( k P  β − . Our code subspace is an image of =1 · ) Y i † n – 23 – i n , α U α , ν i = ) while the total dimension is ) g x n = I R k σ U ν ( 2 2 are Pauli matrices and σ ) ⇔ i 3 are unitary, nilpotent, traceless, and commute with → is , g 2 i , ( i C 1 i g g g σ H j ) in a slightly different form, using two binary vectors auxiliary qubits may act trivially on ⊗ · · · ⊗ ⊗ · · · ⊗ 1 and then act by k , 1 i α . Crucially, the generators i = g ν ) g 3.13 C − + σ x j j can be chosen at will. To describe k ( n H g σ a  i 1 2 (( g = g ±  − j β are chosen to be tensor products of Pauli operators and identity operators ) = · ) can be understood as a restriction on the unitary transformation g = i α ν i as the mutual eigenbasis of the i g is nontrivial, all states in the code subspace will be highly entangled. To  g g( = 1 C 3.13 i U H ) = λ / H . If , is the identity matrix, α, β n generators of the stabilizer group, or U 0 g( errors are correctable, while in fact all σ k The class of quantum error-correcting codes known as additive or stabilizer codes The discussion above applies to the trivial case when the logical qubits are isolated The code described above is degenerate. Different nontrivial combinations of Pauli k − − n . With this definition all properties are automatically satisfied except for commutativity. The coefficient n Commutativity of a pair of generators requires I The form of ( is customary to rewrite ( length stabilizer group acting on individual spins Here on the projected state.from As the a code result of subspace. these We operations will we discuss are later guaranteed whichexploits to errors obtain the can a be idea state corrected outlined in above this way. with the following restriction. The generators of the under correct the error, we can performeigenvalues projective measurements of the from the environment. Noware we subject want to to noise, and consider weto the want recover to at situation use least when them some all to errors.define encode To new that stabilizer end we group perform via a unitary transformation on quotient matrices acting on the differently. The dimension of 2 the auxiliary qubits are correctable. each other, They form an abelian group,called which the acts stabilizer trivially of on We additionally notice that JHEP03(2021)160 ) k as 2 and T α, β z (3.18) (3.19) (3.20) (3.16) (3.17) G . σ H ) = w( c w( ), are elements , by an invertible C H H 3.14 this can be written ) coincide with α, β T ( G understood mod 2, with the binary “parity check” matrix n T from the right by any invertible G × ) G . Multiplying rows of k β. . , · k − . = 0    α    n as a matrix of maximal rank satisfying − ( k k T . 1 − + n 1 − G n β ... ! 2 n β ... β β β I 0 of which span the same space as the rows of , acting on k HgH k + 2 0 mod 2 1 I + 0 k 1 into an − . For binary vectors 2 Z n α ) ... n α ) – 24 – ... ≡ α are to be understood mod 2.

α i − α ) is n    = , β H    α, β i ) = g α 3.15 = g( = ( H g G T . In the example above those would be operators generators of such transformations corresponding to α, β rows, H ) G k k w( 2 α, β + ( rows are generators of logical operations on n commuting with the full stabilizer group are operators acting k matrix 2 H n linear combinations of the rows of 2 1 × − . Assuming algebra over n will have n binary matrix would not change the code, exactly as in the classical case. . All operations with k 2 2 2 i ) T + g Z n k G 2 ∈ + and their linear combinations, understood in the sense of ( c ), there are exactly n ( H × in our example above. These are “logical operations” — they change states from 3.14 ) l k At this point we would like to introduce the quantum Hamming weight We introduce the binary “generator matrix” The operators on ψ acting on individual logical qubits. + , while the remaining x n In the classical caseweight we of all would defineexception the of Hamming the distance trivial ofrows of the one. code In as the the minimal quantum case the situation is more nuanced. The That is why we can always assume that theas first the number ofas qubits follows affected by The similarity with classical codescodewords is striking at this( point. We can identify rows of Its transpose H form ( linearly independent vectors σ binary matrix from thethe left generators would not change the stabilizer groupon but only the choicethe of code subspace into other states in the code subspace. Considering operators of the and introduce a Then the commutativity condition ( is convenient to combine the vectors Because we are working mod 2, the minus sign in front of the second term can be flipped. It JHEP03(2021)160 . d ], . Z C 55 ⊗ H Z (3.23) (3.24) (3.25) (3.21) (3.22) ⊗ . It will Z ]] ⊗ Z . ⊗    n, k, d . Z [[  i i i . ) and qubits. Indeed for any H X 00101 01100 10111 | include many vectors of ..... 1 1 1 10 1 0 0 0 0 2] code was introduced by , with two algebraically- ⊗ l / Im( + i will act identically on T X i − | i i − | 1 . 1) ∈ | i j 3]] G ⊗ i x i , − E g 1 X d , k ⊗ , c / and 00110 01111 11101 would affect strictly fewer than − , summing them up yields a pro- =1 Y k i [[5 n and l X j ..... 11110 0 0 0 01 0 1 1 1 1 + i = [( k i n E 0 = I ⊗ − t | ) i − | i − | i − |    is said to be of type E n 2 2 2 X − Z i − | The X g d = Z ∈ i ( x ~ . E T ) k ∈ 11000 10001 00011 5 − 0 1 10100 n | | 2 0 i − | i − | i − | code. – 25 – , + 0 = ,G i IXZZX ZXIXZ XZZXI XIXZZ , G x, x ]. We use an equivalent representation from [ i 1       3]] 1 2 3 4 , = 54 , 10010 11011 01001 2 g g g g 1 in terms of the computational basis, it suffices to act | | represented by 1 , 01010 l I + g , C | + + − | (1 Z k , H + l [[5 − ) will be satisfied or =1 Y i i n , we only explicitly write the two additional rows linearly , c X ) 0 1 1 00 0 0 1 10 0 0 0 11 1 0 0 0 1 T 3.1 c = , e.g. G 3 P 00000 | their linear combination  1 4 j . Linear combinations of the rows of H = min w( = ,E 1 0 0 10 0 1 0 01 1 0 1 00 0 1 0 1 0 i l d i E       0 | = , . i C H n H 0 | on The code subspace is spanned by two vectors In the literature, stabilizer codes are often specified by writing down stabilizer gener- Since the stabilizer generators are nilpotent, P independent from minimal Hamming weight independent logical operators In terms of the physical basis For the generator matrix The corresponding parity check and generator matrices are ators as products of Pauli matrices, denotedExample: simply as X,Y,Z, perfect andLaflamme, the LMPZ Miquel, identity I. Paz andwhich specifies Zurek the [ code through the following four stabilizers. In practice, to findby states from two such errors qubits and therefore either ( jector on code subspace. Thereforeas the follows: Hamming distance of a quantum stabilizer code is defined A quantum stabilizer codeprotect of against Hamming any distance quantum error affecting at most of the stabilizer group. They do not introduce errors as they do not affect states from the JHEP03(2021)160 code, is the (3.28) (3.26) (3.27) , after ] ] can be ˜ ˜ d d d l i k, k, 1 as a vector | (except the − − ) qubits, where H ]. 1 while the second n, n α, β n, n ] and is reviewed [ [ ( 55 . 2 − T 56 Z d H = , . There is an isomorphism , . = 0 GF(2) = 1) 1) 2 , , ¯ c ,G · (1 (0 GF(4) n we can rewrite can be understood quantum me- invariant. The state 1 c , which is dual to n 2 l ] ). This group of transformations is generated by the Hadamard matrix i ). This class of codes is denoted by ↔ ↔ ≈ 6= 0 Z d + 0 from consideration. z | 2 ] ∈ 1 σ ¯ k, ω c ≈ d 3.14 GF(4) 2.56 · are a special case. They correspond to β + , x ] who reformulated quantum binary stabi- 1 k, is orthogonal to itself. Therefore stabilizer ¯ c n 2 and leave 50 + n Z n, n and i x [ g σ ∈ ⇔ n 2 ]. It suggests a close relation between quantum GF(4) – 26 – n, n Z code corresponds to a classical [ 52 stabilizer codes should be interpreted as quantum GF(4) ∈ code ]] contains no information and one can not speak of ⊂ , called the Gray map, = 0 over ∈ ] α , , 2 2 ). Many other details, including the circuit represen- C ˜ 1 d = 0 ] c Z Gx, x 0) 0) β . k is nontrivial. The quantum Hamming distance , , · k, n, k, d 3.25 = ∈ H ˜ GF(4) 2 d (0 (1 H+ [[ . Then a straightforward check confirms that − and α 4 C n ) are in one-to-one correspondence with self-orthogonal ad- n, n, d [ ψ + ↔ ↔ , c and , which means that the code subspace is one-dimensional. In n, n 2 ) [ 0 3.16 ω c β d · with the scalar product ( GF(4) GF(4) = 0 1 ), ( . Any vector ∈ α k c -th component of 3.14 i . A quantum GF(4) codes of type = min w( GF(4) ] d 2.2 language), and the exchange of n, n, d codes are non-degenerate. They are in one-to-one correspondence with classical [ components, ]] n GF(4) , d in section The equivalence group of classical codes over Self-dual classical codes The formulation of stabilizer codes given above was developed in [ 0 in is the largest quantum Hamming weight of all linear combinations of n, H+ chanically as the group of unitarythat transformations the acting on stabilizer individual generators qubitscalled in remain such the of a Clifford the way group form orinclude ( local permutations Clifford of group (LC). qubits,ω The cyclic generators permutations of of the Clifford Pauli group operators (multiplication by In this case the classical[[ and quantum Hamming distances coincideself-dual and self-dual stabilizer quantum error correction. Rather error detection protocols: theyd can detect anytrivial error one), acting on up to but the relation between smallest Hamming weight of the removing all codewords of stabilizer codes with this case the quantum state where the first equation is understoodequation in is terms over of algebra over codes of the formditive ( codes over 4 By combining the with paper by Calderbank, Rains, Shor andlizer Sloane codes [ as classical additiveunder self-orthogonal addition codes between over obtained by flipping all spinstation in of ( the recovery protocol, can be found inin the the pedagogical textbook review by Nielsen [ stabilizer and codes Chuang and [ classical linear codes. This relation was further developed in a seminal It can be checked that all four generators JHEP03(2021)160 ] ), of 2.19 (3.31) (3.32) (3.33) (3.34) (3.29) (3.30) n, k, d x,y,z [ ) = σ we can ) we can 2.2 3.26 ( t, x, y, z ) ( C . α, β . W  classical codes over y y = ( 2 − 2 − H+ c ] x 4 β, x . · ) , c 50 ( → 1 z , ~ z 2 27 − . ). Enumerator polynomials and ) = )+w 2 c c , z y , of the stabilizer group are real. ( introduced in section ( ) x z z + w 2.58 2 w w z ∈ C x ) − c , ( z y y enables many results developed for clas- , c ]. . β, ) w 0 (mod 2) · c + y 59 ) and ( + 2 ) – c x α g( ≡ 2 y ( + 1) w( 57 1 2 . Using the Gray map j [ 2.57 GF(4) , those whose codewords all have even Hamming + ) − ) = – 27 – n (2 x c → , n x ( H+ II n =1  y 4 α, β Y j we will use the labels of the Pauli matrices ) to ( C ∈C w c X g( GF(4) , their total number is [ ¯ + 1) ω W , y j ) k 3.32 H+ . Then, in addition to the enumerator polynomial ( z ) = − is even, and their total number is α, (2 , ω, ) 4 1 n · c 1 ) can also be defined at the level of quantum stabilizer codes ( n − =0 + 2 1 , but this will not play an important role in what fol- ~ Y z j n ) y c ( x, y, z ) and ( 3.31 ) = 2 z ( ) = + w C c ) + w z x ( c ) ( 3.31 W x ( c ( 1 2 y w y x, y, z ( ). Assuming that the physical qubits are subject to uncorrelated noise, “alphabet” w → ⊥ ¯ y ω ) C c x ) + w ( c ↔ W x ( reduces ( w x ω we introduced even codes GF(4) x y ) c = 2.2 w( ) = w z − c n . In addition to the (total) Hamming weight t w( Real codes make up another class of codes, which is central to our considerations. A Focusing on self-dual codes The connection to classical codes over ∈C c stabilizer code is called real if all generators In section weight. They exist only when Setting the MacWilliams identity ( without any reference to codes over Thus for self-dual codes the refined enumerator polynomial is invariant under OneP can alsolows. Under a definecode duality changes transformation, as follows: the the refined enumerator polynomial full of an enumerator polynomial and we can define the refined enumerator polynomial (REP) introduce weights that count theof number using of the individual “letters”the in corresponding stabilizer the element codeword.write Instead down explicit formulas for the weights these are natural symmetries, which define the group ofsical equivalences codes of to stabilizer be codes. focus applied to on the self-dual quantumGF(4) stabilizer case, and codes, vice or versa. equivalently In on what self-dual follows we mostly (conjugation JHEP03(2021)160 . ) ) ∈ ) ⊂ ⊥ 2 are ) ) z C C z 3.16 . (3.35) (3.36) codes. − Λ( n . This α, β Λ( , σ 2 2 x with the y 2) σ H+ = ( 2.2 )( 4 √ z c n,n / , 3 Z R − generated by ], where it was z ), must also be can similarly be ) x 59 3 + z Γ = ( H+ II are not isomorphic. 3.32 = ( 2 ,W 4 y 2 ⊂ . Refined enumerator . R n ) H+ R of all codewords is even. ,W C 4 + 3 and ) 1 H+ 2 c 4 . Their total number is Λ( ( W y and therefore correspond to ( xz H+ and ⊂ H+ R w 4 P 4 ∗ + 3 H+ Γ H+ II 4 3 by 4 of type x , as described in section , ). The corresponding lattice n C 2 = H+ R 3 is self-orthogonal, the lattice is integral, 4 . ∈ C} GF(4) provides a way to associate a stabilizer C , c ,W + 1) 2 j . If z it is convenient to use ∗ and codes – 28 – (2 ) is self-dual. There is a one-to-one correspondence GF(4) ), the spaces 1 3 C ) + 2 − =0 n,n Y and rewriting its codewords as vectors C j W n 2 (mod 2) R 3.34 y Λ( H+ ⊂ + c, 4 ) = Λ( , while the latter exists for any 2 n n ⊥ 2 x = and have positive integer coefficients. The polynomials C 2) is an Abelian stabilizer group generated by the rows of ( = Λ( , v √ C 2 , we introduce a new version of Construction A for n / 2 Z Z ( . They are polynomials in the ring y ∈ 0) = 1 ⊂ , GF(4) z, W v ) 0 | C , is self-dual, then is a self-orthogonal code from + should be understood as a lattice in Lorentzian space 2 → − C (1 x C Λ( ) √ y C = W v/ . If 1 { ∗ Λ( are refined enumerator polynomials of three particular codes introduced below ). Then the following crucial results follow. The lattice of a dual code . In practice, instead of is purely imaginary, a code is real if and only if ) W 3 C y 6 code to an integral Euclidean lattice in using the Gray map, we define a corresponding lattice using Construction A for σ ) = 3.17 Λ( C ,W ]] n 2 2 2 ⊂ Λ( Z Stabilizer codes are self-orthogonal codes of type Starting from a code of type ) ,W C 1 n, k, d between lattices integral lattices. This correspondence providesof a stabilizer geometric codes. way to Suppose interpret various(or aspects equivalently, The lattice metric ( is equal to theΛ( dual lattice, a self-dual code over C ⊂ binary codes, 3.2 New constructionThe A: Lorentzian connection lattices to[[ classical codes over lattice is not inreason general it self-dual, is even not when connected the in underlying any code obvious way is to self-dual, a and CFT. for To this obtain a self-dual lattice from W in section The rings of invariantdescribed refined explicitly. enumerator polynomials for polynomials of realinvariant self-dual under codes, besides being invariant under ( which satisfy While this formula isThe the former is same defined as for ( even shown that any stabilizerrespect code to has the an Cliffordmodulo equivalent group equivalences, real defined real code. above. codesreal encompass This Equivalence and all is result codes. is defined crucial, with We Since denote because the the it Pauli space shows matrices of that real self-dual codes over This nomenclature was introduced in the context of quantum codes in [ JHEP03(2021)160 , . ) is 1 R Λ ∗ can k ~ ) , 1 is the . The (3.38) (3.39) (3.37) ⊂ 0 L n,n ∗ 2 τ k (Λ ~ 2 d R , / Λ ∗ calculated τ ) ∈ 2 ∗ is the direct ) ) = ( ) R 1 is equal to the ) k ~ C 2 , (Λ α, β , where k = (Λ L . We additionally ) / Λ( , ) ∗ 2 k ∗ 0 ~ 2 − ) being its dual, the = ( C is equal to half the 1 Λ 2 , d πiτ k / ~v 1 = ( ⊥ k 2 Λ( 1 1 d (Λ C − = ~v Λ e ⊃ ) we assumed a diagonal k . As a group, the quotient or ∗ = 7 ≡ ) for min( , or, equivalently 2 2 0 0 E 2 quantum stabilizer code is a 3.38 2 R Λ C p (Λ / , q, q 1 3]] , q , and the quantum code protects − ⊕ β Λ ). The two metrics are related by , (i.e. the number of dimensions of 2 1 1 2 L ) πiτ , − p √ = 3 2 C , 3.17 e α 2 q [[7 . The quantum Hamming distance of d = | , which additionally satisfy are the Construction A lattices of some Λ( β , ∗ v ≡ ) = Λ 2 = | / 2 code and ) · | , acting on the single logical qubit. ∗ C 1 ) 1 v R ) ( | C α = 3 ∗ Λ 3] k ~ C (Λ Λ( , x,z 2 Λ − / 4 σ ∗ k Λ( ⊕ ) 6=0 2 , , q , v C β – 29 – 2 2 [7 , / Λ( ∈ + 2 R The Steane β = 4 v over the root lattice of p 2 2 0 1 7 + q such that α = min √ k E ) = Λ with metric 2 α n d / = C . The number of logical qubits ] 2 L 2) = 2 q p 2 | Λ( n,n , code. √ and the quantum Hamming distance of the CSS stabilizer q 1 v L R | / k ~ Λ 2 ) and in the rest of this paper is that the Lorentzian lattice , d Z ∈ 3]] ⊂ Λ ), 2 ( X v , / , 1 ∗ 1 Λ ⊂ 1 . 3.38 , Λ 2 3.20 ⊥ 2 ) = n, k C ¯ [ Λ τ ). We define the quantum Hamming norm on the space of vectors [[7 ) , is being the Hamming 1 τ, C is automatically integral, is then integral. The number of logical qubits ( via ( C Λ code. In this case 1 ) Λ Λ( C C / ⊂ Θ 4] n,n n,n ∗ , ) n Λ( R R 3 code ) C , Z ]] ∈ ⊂ [7 2 Λ( ) √ n ( 2 logical qubit against any one-qubit errors. Geometrically, 2) α, β n, k, d , which corresponds to Pauli operators A Lorentzian lattice A particular class of stabilizer codes, the Calderbank-Shor-Steane (CSS) codes, can [[ 2 √ / Z = 1 = ( Z where we use different letters (Greekthe or form Latin) and of a the vector metric arrow (or tensor. lack thereof) to imply which is a holomorphic functionimplicit of assumption two in complex ( variables is simultaneously equipped withLorentzian the metric, while Euclidean previously metric. itthe was following In transformation given which ( by preserves ( the Euclidean metric on be characterized by the Siegel theta-function CSS code with Hamming k the quotient of the weight latticeis of dimension of the torus code is the smallestwith the length Euclidean of metric. anyHamming It distance nontrivial is of vector in in any case not smallerExample: than Steane easily be understood geometrically.sum In of this two case EuclideanFurthermore, the lattices Lorentzian lattice require classical binary codes where the norm on theelements in quotient is the understood preimage. to be the minimal value of the norm on all number of generators inthe the torus abelian group the ( ~v JHEP03(2021)160 . , , ) 0 /τ can . +1 6.6 1 , τ τ, τ τ ( (3.42) (3.43) (3.44) (3.40) (3.41) GF(4) − ) C → 6.2 = → − Λ( τ 0 GF(4) . Θ τ ) τ 0 τ ( over codes discussed in C . 0 a, b, c τ = , , with respect to which 0 , is isodual. The theta ) ) 2 GF(4) satisfies τ c and , c y ( ( 0 , we immediately conclude ) 1 z Hamming weight, which is τ n . C , b + φ 2 ) 0 associated with the stabilizer 0 2 . R ) Λ( 2 x → C ). The invariance of y ) + w τ, τ 0 easily follows from the invariance c ( . Indeed, it is straightforward to Λ( upon the substitution ( GF(4) 2 0 Λ 3 y , a + 3 a a 2.33 ) q 0 Θ ,
, 2 /τ 0 ) 0 2 1 x = Z τ, τ , the n/ c c will turn out to count dimensions of the 0 may also be considered. It is associated = ( ) ) , a a and the new Lorentzian Construction A 0 n, q ) + 2w 0 C ) 2 − 2 c β → − τ 0 ( Θ y · ττ c c ~x,~y ¯ x τ − 2.2 b b α + , or , − GL(2 τ, τ 0 2 associated with the Hamming weight of a ( − – 30 – of the Lorentzian 3 /τ ) = ( x ∈ C 0 2 ) = ( 2 1 ) = w − y Λ , b b Θ β c Λ 0 ) correspondingly. , /τ ( ) g 1 = b + α, β τ c c via the Gray map is interpreted as an isodual binary +3 T → − d 0 ( − 2 2 0 3.32 τ Λ C + τ α x φ = ( 0 /τ, = 1 ~v group acting simultaneously on b b → 2 q and − and ( | ) ). We postpone giving explicit examples until sections 0 code ( C v Z | y Λ , c c W are defined in the text after ( + 1 Θ 2.62 n 2 0 + − τ 0 → − GF(4) b b y PSL(2 → a, b, c , the Siegel theta function of a lattice 0 ) = 2 τ ¯ τ , 2.2 τ, under ( ) + 1 . When it is self-dual (unimodular) it is also covariant under ) C , changing as and τ ) can be interpreted as an orthogonal rotation of 0 were given in ( Λ( , understood as a Euclidean lattice with metric ) upon the substitution 1 + 1 ) , Θ → /τ 2.1 0 0 C is determined in terms of its refined enumerator polynomial 1 3.17 x, y, z τ τ φ ( ( 2.55 C Λ( C g Looking ahead, the theta function Besides the Euclidean metric Comparing Construction A from section Similarly to the Euclidean lattices associated with binary and When the lattice is even, it is trivially invariant under the simultaneous shift → W → − 0 0 and that function of this Euclidean lattice follows from CFT operators. In full analogy with the case of classical binary codes, for which we defined with the binary Hamming distance the original self-dual code. Given that the generator matrix where code, the conventional Euclidean metric Therefore the Siegel thetatice function ( will becomecheck the using theta Jacobi theta function function of identities the that Euclidean in this lat- case defined above, we findbe that associated a with classical both(self-dual) self-orthogonal a lattice. (self-dual) Euclidean code These integral Euclideanterms (integral) and of lattice Lorentzian the and lattices Euclidean athe are vector same Lorentzian related as integral to the each quantum other. Hamming weight, In is The theta functions under of antly under the full sections code τ τ Thus, the Siegel theta-function of an even self-dual Lorentzian lattice transforms covari- JHEP03(2021)160 ) . is + 0) ) ) 0 x 0 ν, ( (4.1) 3.46 to be , pro- (3.46) (3.45) ψ ) ∓ 2 will be 0 τ, τ , which , ica ]. 0 δ ( ) ) µ ) ) changes C C C ac ) = τ, τ ± 61 ( ( , 0 ( , the scalar 0 x − ) Λ( ) ( Λ ν, is also even, , Λ C , 60 0 3.46 C ( ψ Θ 0 R + C Λ ac Λ( necessarily have ( µ starting from the Θ C ( is the momentum. ) 0 , expression ( 2 ∈ , C C Λ x W 2 n/ W ) 0 Λ( − √ = ˙ ∈ C a, b, c / ) ) p 0 ) ττ ) = → , ba 0 α, β 0 α, β ν, µ ) = ( , c 0 0 ± = ( , ab should be odd. Provided both = ( , where 0 , we can take the same vector , b /τ must be periodic, t 0 ) ν, 1 ( ~u . The procedure is identical to the a ) ab C 2 C ( − x δ (0) C , c ( Λ( W p ) ψ +1 W c /τ, mod 4 ( n is real, i.e. it is invariant under complex 1 z . 2 if and only if the code n ) ) = is odd, which is the condition for − 2 0 (0)+ ( 2 ˙ x C ) 0) + x 2 C , 0 ( µ, ν is divisible by four. Similarly, ( τ, τ 0 dt ) + w ( c Λ ± bc ) n ) = ( , resulting in operators of conformal dimension C Z mod 4 t – 31 – Θ x ( − 2 µ, ( 0 0 ( w n Λ x = √ with odd norm C / Θ , cb to be odd, S ) = 0 when Λ W 1 . Upon substituting 2 0 cb − β δ ∈ τ n · + +1 2 δ ~ is even, and accordingly the theta function 0 0 1 ~ 0 2 ↔ ) τ , bc 2 C + ( τ ( 0 → C α , with the only change that scalar product is now understood 0 Λ · for a real self-dual W τ 1 ~ ) ) = (2 , + 2.1 C ) = C +1 Λ( α, β ~u τ Λ( . To partially resolve this problem we employ the procedure of twisting ( · Θ ∈ δ ~ → n , given a vector 2 2 δ ). ~ gives τ . Additionally, for 2 = 1 Λ 2 / δ ~ as in the binary case. Notice that for any ) H+ R 2.40 ) = 2 R 4 . From here it follows that 0 2) p ∩ √ + is also even. When additionally τ, τ µ, ν 2 L ( (2 ) C p H+ II / C 4 1 ( As a warm-up we consider a particle of unit mass on a circle of radius In the context of Construction A code lattices ~ 0 Λ = Θ The classical EOM can be easilyAt solved: the quantum mechanical level, the wavefunction 4 Codes and CFTs 4.1 Narain CFTs We start with a briefspace review of of Narain toroidal CFTs. compactifications This of topic string is theory discussed and in the many moduli textbooks including [ vided odd, the Lorentzian lattice invariant under reduces to ( When the code isfor self-dual any and real conjugation, accompanied by covariantly under modular transformation C ∈ of these conditions aregiven satisfied, by the Siegel theta-function of the new lattice product with and therefore by a half-vector, which will yieldoriginal a lattice new even self-dual Lorentzian lattice one described in section to be defined by the Lorentzian metric. δ also want to maximize thedefines Euclidean the norm of spectral the gap shortestbinary of nontrivial codes vector the also of theory. appearsvectors The here: of same the construction problem form A we∆ Lorentzian = encountered lattices ( in the case of a lattice and sought to maximize the norm of its shortest vector, in the quantum case we ~ JHEP03(2021)160 . , . , t Λ Γ ~ V are n,n ) = ∈ (4.9) (4.4) (4.5) (4.6) (4.7) (4.8) (4.2) (4.3) (0) π R p πR π~e in 2 = + 2 2 / ) stands for + 2 Λ should now R , such that (0) + ˆ x Γ t, σ ∗ ) ~p . x ( t π ∗ Γ ( , ~ 2 + Γ X ) , and operators x and σ ∈ being quantized, ) = ˆ L ∈ , . t − , in which case x ) ~p ) t ˆ ( p ( ~ ( σ σ ~ P n/R ˆ 0 P x + − in . α t . t = 2 ( ( − J  , 0 e e in in J Γ 0 α n − − n b IJ forms a lattice e e X ∈ B n I n ∗ + n n b ˙ a Γ , while ) X − σ is a lattice and Γ . I 6=0 6=0 , ~e ∈ IJ + Z X t X n n n ~e V . The solution for ( must be physically equivalent for B ∈ 2 ~ ˆ i i p 2 P 2 R 2 I) in , = ~e ∈ − π~e , Γ with eigenvalues − − ⊂ e  ) ~e ) + ) + 2 J n 0 · 0 σ 0 ∈ B Γ n σ σ a α (0) ~ X P X ~e 2 = − is periodic, and therefore ˆ ) + 2 p − 0 + + ( / t 6=0 4 − t IJ t ) α ( X σ n ( ( ~ 2 P R t, σ B R as a sum of left and right-moving components, 0 i 2 ( L R ˙ ~p = 2 2 ~ ) α X X ~p ~p − ~ , where 0 0 – 32 – X  2 R + − I α α p = t, σ Γ) ˙ L ) + ( X dσ π R + + − ~eσ is only defined up to an arbitrary shift by ~p σ  ~ and π ( X 0 (2 ) 2 2 L + is not well-defined. Since the points ) + 0) 0) 0 / -field does not enter into the EOM, nor into the solution p α dσ , , Z n t ˆ x B 2 2 ( t, σ π for all possible ≡ R ~ (0 (0 V t t, σ 2 ( L , ~p dt ( 0 2 ) ~ ~ ~ | X X X Z ~e X ~ R X ~v Z | π 0 1 0) + , ~p . 2 , , ) must be a vector from the dual lattice, + I) L Z ) = ) = ) = 1 ~p π πα (0 σ σ = 0 now stands for the eigenvalue of B 2 4 ∈ 4.4 ~ α I t, σ X p + − ( = ( on a torus P = m t t + ( 0 ~ ( ( , X I ~v α S ~ L ) = R P ), we can represent dimensionless, we use the conventional choice 0 X ~ ~ X X α where R t, σ 4.3 m/R , and ( Z = , ~p ~ . The antisymmetric X = L ∈ L B~e . The worldsheet spatial variable ~p ~p k π~e bosons n + Γ , n ~ P for ∈ 0 )+2 , which results in the quantum-mechanical momentum operator α ~e ) Next let us consider a two-dimensional classical worldsheet theory describing the mo- n/R (0) t, σ ˆ x = ( = πR ik ~ ~ The set of vectors To render becomes even and self-dual. To verify first property we calculate Quantum mechanically, since the total momentumV ( At the classical level, the vector Going back to ( but it affects the relation between the center-of-mass velocity and the total momentum As in the previousany example, X tion of that lattice rescaled by p be understood as thewith solution the of caveat the that EOM theequivalent, in operator the the algebra Heisenberg picture, ofe operators consists of 2 JHEP03(2021)160 Λ . For (4.14) (4.15) (4.16) (4.10) (4.11) (4.12) (4.13) Λ ∈ δ ~ 2 . ¯ τ is πi 2 τ − e = . ¯ q , and the vectors of the Λ . . ) correspondingly. Then it Z m R ∈ ). We start with a vector , the B-field is trivial. We , ∗ p ) ∈ 2 Γ X 4.10 R − πiτ √ . 2.39 2 , ~p L e p L 2 and CFT are indexed by elements of compactified on a circle of radius ~p = The simplest example of a theory mR defines an even self-dual lattice ( were two independent holomorphic Γ , = d ) 0 , Z X − τ 2 Z , n, m ∈ n ). In this representation, and taking R √ 1 d, U(1) , , / , β 1 : for any integer , q and = R × ) 2 ! R 1 z τ 3.17 / p n, m d (¯ R , which automatically satisfies 2 2 R ∈ R GL(2 γ R p + Bγ ~ √ X ¯ ∈ q ! ∈ · a, b L ∗ in the coordinates ( 2 R , p U(1) – 33 – p 2 g / 0 γ i~p 2 L 2 is modular invariant and the CFT is well-defined. Λ √ mR 2 p mR = = ), where )+ ) ,

mR q z ¯ τ ( 2 = Λ Λ L = + g ∈ ~e √ τ, R ~ 3.38 ) X is ( T · n Λ n R R L Z Λ Λ

X for all possible , ~p , α i~p ) . Since there is only one boson = L e ) ~p 1 ( R L are related by complex conjugation. R ) = p d α, β = : , ~p 2 ¯ are the generator matrices of τ ∈ | L ) R T 1 ~p τ = ( n, m with some integers ) ,p ( ( 1 L v and η v p n/R | as follows − = ( 2) V τ γ ) ~v √ = ) R consists of points ) = (2 is self-dual. ~ = ( P , ~p ¯ / τ Γ ) ∗ Λ L 4.14 τ, γ ~p , ( Λ Z = ( -coordinates the metric is given by ( and ) ~v . It is much simpler to represent γ a/R, bR , the generator matrix of 1 , is an even self-dual lattice, 1 The primary vertex operators of the α, β ( = 2 R Λ = (2 . The lattice 0 We would like to applyδ to this lattice the twist procedure ( immediately find The set of vectors in Example: twist of aof compact the boson type described on aboveR a consists circle. of a singledual boson lattice are If We emphasize that in constrast to ( variables, in ( the lattice The partition function of this theory on the Euclidean worldsheet torus is straightforward to check that and therefore where vectors In α To verify self-duality it is convenient to perform a linear change of variables and represent JHEP03(2021)160 = 0 , or R Z ) and (4.21) (4.18) (4.19) (4.20) (4.22) (4.17) ) such ∈ for any satisfying 4.14 R related by 4.10 F k is even, but 0 Λ , n, m . , ) = 2 } } + 1) 0 Z m Z and 2 k R matrices ∈ , ∈ Λ 2 d 2 . They form a group n/ × ( d = 2(2 v n, m defines a CFT, with the d R a | 2 ) ∈ n, m, k, l d,d | R , m R 2) ~p . / ⊂ } + 1 , Z ) Λ n + 1 Z and ∈ is odd. Then (2 , l d v has an obvious symmetry: any element , or the other way around. In the first { R I d, d, Z + 1 n, m , = + 1) | ∈ I ∈ O( k as the union of two sublattices ) 1 . Therefore either is the generator matrix of a particular even and applying the twist procedure repeatedly, 2) O } L Λ (2 × , m I ~p v Z d, d ) k, l R = – 34 – d , ∈ + 1 are related to each other by boost transformations O( } Λ ) + ) is manifestly invariant under orthogonal trans- n, m/ O( n Z d,d (2 (2 × , m ∈ R Therefore the full Narain moduli space is given by v n, m v is defined as a group of integer ) 4.14 | is odd, { · ) d 2 ) ⊂ 2 mod 2 = 1 ]. + 1 ). CFTs of this kind are called Narain CFTs. A central gives δ Z = 2 O( Λ is given by all vectors of the form n 0 1 0 n, m + 1 62 ab/ | 0 n, m , Λ Λ l (2 ( 4.14 ) d, d, ), Λ ( v v , 25 { 1 { O = 2 Λ and = n, m 3.17 b yields another CFT of the same type. In fact, this construction 0 ∪ δ . The lattice generated by (2 Λ = Λ 0 δ v + d,d { is even and and the identity matrix 1 R is the lattice of the single boson compactified on a circle of radius ) ) acting independently on = 0 transformation are equivalent. of symmetries of the CFT, which is a subgroup of T-duality transforma- maps it into itself. 0 ) Λ Λ = Λ = Λ ) d d, d Λ ) R n, m d 0 1 ) Z Λ O( (2 O( d O( v . At the level of the generator matrix, and working in coordinates ( · . O( ) . × . Thus, starting from some radius g d, d, δ k ( ) O ∈ × 2 R = d d, d O L As we will see shortly, any even self-dual lattice The CFT partition function ( In the example above, it was obvious that starting from a toroidal CFT ( F . In the second scenario ) Here and in what follows O( = 2 2 O( to be odd, we must require g d 2 0 T 2 from that metric is given by ( where self-dual lattice in partition function given bymathematical ( result, which provides a descriptionis of that the all moduli even space self-dual of lattices allin Narain theories, formations O( tions. Thus, from thean CFT point of view, two even self-dual lattices integer twisting by a vector can be applied to generaldegrees CFTs of of freedom this as type, well and [ can be extended to CFTs with fermionic In other words R/ R we arrive at the lattice of a boson compactified on a circle of radius The disjoint union of F where not doubly-even, while scenario we represent δ JHEP03(2021)160 , , ) ) ) C i ) i x C Z n σ Λ( n, O( × O (4.23) . I of O( ) × and ) can be q ) i C z (¯ ∈ n 4 σ Θ p θ 4.21 O( O = -th component i ¯ c , transformation. We ) q ) (¯ d 3 represented by θ (exchange of ) O( ¯ ω = n × b ¯ ) ↔ O( of T-duality transformations. d ω ) , × ) ) cover all possible Narain CFTs. are in one-to-one correspondence n O( q ) (¯ C n 2 O( 4.8 θ O( × . In the previous section we saw that = ) and by exchanges of the ¯ a n,n n β R O( , ) ⊂ q and ) -th component -th coordinate. In other words, the subgroup ( , i i 4 C – 35 – . α  θ ¯ a Λ( C = that this is not the case, and therefore any pair of ⊂ ¯ c, a c n D . Using T-equivalences we can bring the code generator 2 defines a Narain CFT. We therefore arrive at the main , c n ) − ) β 2 | q Z b ¯ ) ( is given in terms of the refined enumerator polynomial of 2 n,n τ 3 is also an element of ( θ R C √ ¯ c, b ( η i x | Θ c ) by means of an appropriate = σ n 2 + 4.8 . The permutations are orthogonal transformations ) define a family of Narain CFTs, which we will call code theories. b ¯ n , which are not T-equivalences. Therefore, two code CFTs associ- b and , b flips the sign of the i x representation of codewords, T-equivalences are generated by simulta- )  R H+ σ ≤ ) i q ) z 4 C , where ( σ i Z n 2 2 W θ → | α, β n, ≤ ) -th component of i z ) we established that real self-dual codes i τ = 1 σ ( O( = ( ) = η a 3.2 | ¯ τ c ∈ → τ, ), i i y ( O σ Z with the 3.41 In the It is important to ask whether any other T-duality transformations from The code equivalence group (Clifford group) includes arbitrary permutations of code- Thus, CFTs with momentum lattices of the form ( → α i x alent code. WeT-dual show code in theories appendix are equivalent also in the codeneous sense. permutations of the componentsof of be immediately notedσ that the fullated equivalence with group equivalent also codes includes arephysical not properties. cyclic necessarily We permutations equivalent will as see CFTs many and such may examples have below. besides different those mentioned above, can map a code theory into a theory based on an inequiv- The exchange of where of the equivalence grouptransformations, generated which by leave these alllows transformations we physical will is properties simply a refer of to subgroup the these of equivalence CFT transformations T-duality invariant. as T-equivalences. In It what should fol- word components and conjugations offor the arbitrary which are diagonally embedded in the group via ( any even self-dual latticepoint in of this paper:dual real codes self-dual of quantum type stabilizerThe codes partition (or function alternatively of classicaldivided a self- by code theory is given by the Siegel theta-function brought into the formdemonstrate ( this explicitly in appendix 4.2 Code CFTs In section ( with even self-dual lattices dimensional CFT. Thephysically first equivalent two lattices factors to insymmetry each the of other. denominator a particular act The lattice, from which last the maps factor left different acts lattice andIn points from relate other into the words, each any right. other. Lorentzian even It self-dual is lattice a with generator matrix ( where the denominator represents the group of T-dualities — symmetries of the two- JHEP03(2021)160 n × n (4.29) (4.26) (4.27) (4.28) (4.24) (4.25) ) into an identity ) is rows to eliminate ). After exchanging m , and therefore they 4.24 4.26 i β − 4.25 n ), up to multiplication from    to bring the generator matrix ) . This is the “canonical” form m β  4.26 − ) B) n | m ( and − × , a ) n ( 2 α m 2 × = ( I / − √ ) . ) and arbitrary . n / 1) T m ( − 0 − I) ! G n n | ( 4.24 ( I I n m B , mod 2 , and using the last ). For a real self-dual code the binary matrix × i m ) using additional equivalence transformations, T ) – 36 – β = 2 a B × rows, we finally transform ( is equivalent to a B-form code. This result has m = ( B 0 ]. 2 I j m − − 2.15 I T 2 , 4.26 n 63 m

( 1 n H+ binary matrix, B-form codes. There are (compare G B = 0 with 4 − × =0 can be brought to the form ( = given by ( ) Y j n n i associated with the B-form code (    m i T Λ α ) − × binary matrix. The codewords which form the rows of α = G C n ( ) n 0    Λ(  m 1 n with α α ... − to the form )    i n rows of i ( α , β i m × , such that α ( } − m components of 1 n ± ) , components of the first m 0 ) to the following simple form. (We also need to perform linear operations m − ∈ { )) − n is some ( 3.19 ij n a B 3.35 Since any self-dual code is equivalent to a real code, and the generator matrix of any The generator matrix of has zeros on the diagonal and is symmetric. Otherwise it is arbitrary. For notational where real code can bewe brought to conclude the that form anybeen ( code established of in a type different way in [ with ( distinct B-form codes in total, and any real code has at least one T-equivalent B-form code. We will call codes whosethe generator left matrix is by of a the non-degenerate form ( the last matrix, yielding a generator matrixof of the the generator form matrix,B analogous to ( convenience, we prefer to exchange allto components the of form are orthogonal to belong to this code. Inmod other 2, words, by the an last the appropriate linear last transformation in the algebra where the matrix mod 2 with theor codewords the that lattice.) change the First,matrix by generator using formed matrix, linear by but operations the do and not permutations, change we the can code bring the matrix ( JHEP03(2021)160 , p ) by . O Z n,n (4.36) (4.32) (4.33) (4.34) (4.35) (4.30) (4.31) = 0 R n, n, B , O( ) Z . . within ) ) Z n Z 2 . n, n, -th element, which 2) , O( ii ) , which we denote n, n, √ n, n, ) Z / ∈ Z O( Z O( n, ( ! ∈ ∈ ii 1 n, n, mod 2 = 0 ! ! ii GL(2 1 I − 2 B B O( / I D ⊂ B I 0 : ) . ii ) 2 1 Z

-field flux, as well as their T-duals. Z ii C , A 1 ) √ B 2 − = / Z I

n, n, follows from F ! n, n,

. ) = I ) 0 ) O( 0 O( n, n, are related by T-duality transformations in Z → Z , but any choice results in the same lattice. ∈ H i ij B , 0 can be obtained by the action of FH O( 2 I ( 2 O ! B ) related in this way, the lattice remains the same. ∈ n, n, )) which preserve the lattice. – 37 –

Λ √ n, n, I C ( X / 2 Λ ) can be obtained from the matrix with ! , F 0 O( Λ( ! O I 0 B 4.22 , D with quantized and H I ) 0 4.28

) ) = Z π ! C A 2 F F = 0 I 0 ), we find that the code theories are toroidal compactifi- ( 2 I ( ,

) . For all G Λ Λ

n, n, Z T . All such generator matrices are related by = 0 may result in the same lattice, defining an equivalence class 4.11 2 I F ji X O( ) H

B − Z ∈ , n, n, = 2 = G = , ! O( √ Λ ! p → − n, n, / X ) with ( 0 I O 0 , ji form a congruence subgroup within ! → } to be antisymmetric, reducing the sign ambiguity to the simultaneous O( p B defines an even self-dual lattice, a sublattice of 1 I 0 4.28 B : Λ , , are the following elements of 0 ij O ) H ) ± 2 I i ij , F ∼ F H B Z F

O F ∈

0 0 ( B 2 I Λ H → -fields corresponding to the same

. Accordingly, all real codes can be described as a coset (from the denominator of ( ∈ { , ) n, n, is a diagonal matrix with all elements being zero, except for p B ) → − Z F ij O ii Z O( ij 1 X B Different The T-duality transformations which map a code to another code, permutations Comparing ( n, n, . Finally, the generator matrix ( ( n, n, 1 2 Such matrices O The equivalence of the lattices For any within Therefore all Construction Awith the lattices generator matrix being where is a transformation from and sign flips where cations on the cube ofDifferent “unit” size O( Let us choose flips There is an ambiguity is choosing signs of JHEP03(2021)160 , 0 )B n (4.40) (4.41) (4.42) (4.37) (4.38) (4.39) ) with 4.31 ( , . . . , β i +1 . Equivalence O m B , , β    m 12 , we have introduced b } 1 1 , 11 − 12 . , . . . , α 0 b i b 1 , 1 β α 12 T 1 i − 11 , ) of a code theory does not b . ) b ∈ { − α α , ( i − − is again a transformation of the i y B α ) = ( 6.2 4.39 X w it can be written as + I) 2 22 n ), but there is a particular set of − b ) is the analog for code theories of ) D = B D α y 1 and 4.28 w( 11 − 1 w , 4.36 ( . b z and − 11 T p ) B + , . . . , α ) 6.1 b ) 12 T α 2 1 Λ O 2 ( b D +1 y Z D Z ) as the Schur complement, with the property B m , w    )( y p y ) , α D w = α – 38 – 4.39 n, n, 0 n, n, m , and ( − w( 2 B → O α ) − O( in ( i O associated with n B β is the diagonal matrix with ones in the diagonal en- x 22 ) by reducing mod 2. We illustrate the coset construc- C b = B , . . . , β i + → ). + I)B + 1 D α i ) and zeros elsewhere. We note that the composition of X β β codes in sections D α , consistent with the consecutive action of 4.37 ( ( 2 = 4.22 ((    i 4.39 via D C X = 2 i 22 12 → + W b β b ) parameterized by n in ( 1 B The T-duality transformation ( includes all binary orthogonal matrices with zero satisfies w = D 11 11 T 12 0 4.40 b b 2 b and = B O    D = 1 n B = ) are mapped into ( ) with is an arbitrary nondegenerate submatrix. It can be written as . i 4.36 11 b 4.40 Most T-duality transformations (we only discuss those which map code theories to There is a similar coset construction for all self-dual codes, i.e. including non-real codes We recognize the transformation of that the new matrix where the sum goesauxiliary binary over variables all values of binary variables form ( different Consistency check. change the CFT partitionpolynomial function, invariant. and For the therefore code it should leave the refined enumerator where all algebra istries mod associated 2, with and two transformations ( where transformations which do. They include permutations of and “genuine” T-duality transformations Here the subgroup classes ( tion in case of code theories) do not preserve the B-form of the full Narain moduli space ( (odd self-dual lattices), The coset description of the real self-dual codes ( JHEP03(2021)160 0 B con- → (4.45) (4.46) (4.43) (4.44) j ]. That B codes with 65 , ]] 64 . j α n, k, d invariant under the [[ ij C B i ψ i, it can be written as α ]. Returning to self-dual 6= B is a new graph defined as i>j 64 X , is a new graph defined as i j. i ∗ ∗ ∗ ) = ) i i B B maps B-form codes into canonical α (we associate the graph with the ∗ ( x , k, l ) σ j B . ∗ ij ) are called graph codes [ . In terms of B i and ) , f mod 2 j z i z ) is simply the graph isomorphism which n σ σ , denoted 4.43 kl ( i , defined as the state = ((B )), δ as a subgraph consisting of all nodes C i n ik 4.38 =1 . In other words, the transformation i Y j H ∗ – 39 – m 3.22 i x ) , . . . , α σ j + B 1 × ∗ il α = with zeros on the diagonal can be interpreted as the | ) ) remains invariant. m — as a repeated application of local complementation B ) i i (see ( i , and the edges between them. The complementation g ]. B ) ik ]. Many aspects of code theory, including the action of α ∗ i nodes. In this way all B-form codes (and code theories) ]. An alternative language, also used in the literature, is is ( f 4.42 70 n [ i, j 68 = 1 11 ( 69 ) describes all possible compositions of edge local comple- , 1) + B ((B ) , b . Local complementation i − ij kl 67 α 63 ( for all → ( 1 B B 4.40 , f C B , while ( =0 → ψ X i α m kl = n 1 B ≤ C 2 + 1 mod 2 i ψ remain unchanged. Edge local complementation is defined with respect i kl = ) is in a nutshell the statement that any stabilizer code is equivalent to a g ij B C for ]. We should also mention that non-self-dual codes, i.e. B ]. Local complementation of a graph ψ i 4.43 66 β → 71 , i.e. , i.e. such that i i and g ↔ kl i ii B α B , also can be represented as graphs with labeled nodes [ 0 In terms of graphs, the permutation ( The binary symmetric matrix to an edge — a pair of vertices Two graphs related tomentation each are other said by a to sequence be of edge isomorphisms local and equivalent. edge local The comple- edge local complementation (ELC) simply complementation applied to the neighborhood of while adjacency matrix) with respectfollows. to We the define node thenected “neighborhood” to of procedure, applied to anodes (sub)graph, which were removes not all previously existing connected. links, and At connects the level all of pairs the of adjacency matrix this is the context of graph statesthat [ of boolean functions relabels the nodes, whilementation ( [ is the so-called graphthe state equivalence [ group (Clifford transformations), have been discussed in the literature in transformations from the Cliffordof group) the to form the ( canonicalgraph form code with [ stabilizer generators k > codes, the one-dimensional codeaction subspace of Stabilizer codes with generatorswe of can the always form bring ( a stabilizer code by means of equivalence transformations (unitary swaps adjacency matrix of a graphcorrespond uniquely on to graphs. Exchangingform, all and the stabilizer generators in this case are where we have assumed that JHEP03(2021)160 , 3 ELC n t (4.47) ]. The (multi- 71 , x,y,z 70 σ . ] defined via )) 72 . of a binary matrix X , ) 71 ELC n . X i 1) − Ker( , x (1 Q ) := dim(Ker( ) = X ( x nodes is known as the OEIS integer ( q n is related to the full number of equiva- . The kernel on w , s ]) ∈ ELC n . w i ELC n t 6 i, j (B[ s – 40 – y ]) for w ij codes) are not T-dualities. For simplicity we consider (B[ B s − n x GF(4) . As the number of inequivalent graphs grows rapidly, it is } 1 ,...,n 1 X ⊆{ w ) = , see table via the Euler transform. (We note that the code CFT for a decomposable x, y ( ELC n 1 1 2 4 10 35 134 777 6702 104825 3370317 231557290 1 2 4 9 21 64 218 1068 8038 114188 3493965 235176097 1 2 3 4 5 6 7 8 9 10 11 12 t in the language of Q denotes the submatrix of ω A156801 . Number of ELC equivalence classes of indecomposable graphs ] n . Number of equivalence classes of graphs under edge local complementation (ELC) w 12 ELC n ELC n i t ] as a means to classify equivalence classes of classical binary codes. Their connection B[ ≤ In the beginning of this section we mentioned that cyclic permutations of Since any code theory can be brought to the B-form using T-duality, and the interlace As was mentioned above, T-duality leaves the refined enumerator polynomial invariant; To summarize, B-form codes (graph codes) are in one to one correspondence with is the same for all codes associated with edge local equivalent graphs. There is another The conventional definition is related to our definition via n 21 3 C interlace polynomial is ainterlace proper polynomial characteristic will of be the given code in CFT. section Explicit examplesplying by of the Here is understood to be with respect to modpolynomial 2 algebra. would be the same no matter which B-form representative we choose, the W homogeneous polynomial with this same property,to i.e. it the is same the ELC same for equivalence all class. graphs belonging This is the interlace polynomial [ equivalences. If we consider theirlevel action restricted of to graphs, the T-dualities space of areand B-form generated edge codes, by then local permutations at the ofB-form complementations. nodes codes) (graph are In isomorphisms) T-equivalent if what they follows belong to we the will same simply ELC equivalence say class. that graphs (or lence classes code is the tensor product ofthe the original CFTs associated code with the factors.) indecomposable codes into which graphs, and we willa use code both theory languages into another interchangeably. code The theory T-dualities are that necessarily code transform equivalences; we call them T- with stabilizer codes and Cliffordnumber of transformations classes has of also ELC beensequence equivalent discussed graphs in [ more convenient to keep trackedge of local indecomposable equivalent graphs/codes. indecomposable graphs The number of classes of equivalence classes of graphsof are physically therefore equivalent B-form in code one-to-oneELC theories, equivalence correspondence i.e. classes (which those with are related defined the toin to classes include each [ isomorphic other graphs) by have T-duality. been studied Table 1 for JHEP03(2021)160 . 8 is 12 z ≤ σ ≤ n (4.48) . Pro- n ) ↔ C x Λ( , for σ LC n t and exchanges . E . x,y,z can be written as , the new lattice will be self-dual σ ) LC n ) i C , ), one can occasionally find transfor- ], in particular to classify equivalence ! Λ( α, β ( 75 – 0 1 1 1 4.31 ( on

i 73 , ) ω O nodes is known as the OEIS integer sequence 69 , α, β n – 41 – ( 34 ]. Two graphs related to each other by a sequence on → 73 ) , LC n t 69 α, β , ( generated by . C-equivalent codes necessarily share the same enumerator 63 . The action of β ω )[ x,y,z σ and ]. 4.46 α 75 ,[ 2 1 1 1 2 4 11 26 101 440 3132 40457 1274068 1 2 3 6 11 26 59 182 675 3990 45144 1323363 1 2 3 4 5 6 7 8 9 10 11 12 n LC n LC n i t , see table . Number of classes of graphs equivalent under local complementation (LC) and multiplication by -th components of The relation between quantum stabilizer codes and 2d CFTs outlined in this section The role played by ELC graph equivalence in determining the physical equivalence of If two codes are equivalent in the code equivalence sense, but not related by T- i = 1 not previously been performed.nodes, We obtained provide with a help full of classification computer for algebra, graphs in on appendix is up only to one particular aspectcodes play of in what the is context likely of a chiral much CFTs, richer we story. can essentially Given take the for role granted classical that quantum polynomial but usually have different refinedC-equivalent enumerators. codes The are code generally CFTsB-form associated physically codes with distinct. correspond At to the the graphs level related of by graphs, LC,the C-equivalent but corresponding not code by theories ELC. motivatesof us graphs to that classify correspond ELC to classes within equivalent codes. the LC To classes our knowledge such a classification has of classes of LC-equivalent graphs A094927 equivalence (T-duality transformations), we callfor them cyclic “C-equivalent,” permutations where of the C stands tation equivalence classes of graphsof are therefore equivalent in codes, one-to-one i.e. correspondencelocal those with Clifford classes related group to in each theisomorphic literature). other graphs) have by LC been the equivalence studied classes in Cliffordclasses (which [ group are of defined (also self-dual to called quantum include the stabilizer codes (or, equivalently, graph states). The number mations which transform a B-formcodes code related into another to B-form a code.equivalent, at given The the one orbit level of via of allcomplementation cyclic graphs, B-form (LC) permutations ( to of the orbitof with isomorphisms respect and to local consecutive complementations actions are of called local LC equivalent. Local complemen- where algebra is modvided 2. that This we start action with automaticallybut an extends may even not to self-dual be Lorentzian the even. codecodes Thus, lattice being cyclic permutations, real. in Combining general, cyclic dothe permutations not of preserve different the components property with of exchanges of n Table 2 Number of LC equivalence classes of indecomposable graphs JHEP03(2021)160 ), to ] it ) is 270 76 3.15 (4.50) (4.51) (4.52) (4.49) 3.14 GF(4) out of satisfy this in ( 3 ) , c 103 (  , , = 2 because of ( as generators may i 2 n , = 4 )) + c . g c ], which establishes a n 2) ( n mod 4 ] uses a code equivalent c z ↔ 76 1 w 76 1 + , , α c y x 1 − 2 c GF(4) σ σ ) ([ ) β r p 2 i ]. The expression for the code i ∈ c ( 2 76 4.44 z → → ], has chiral vertex operators of , c ) = g( )) = 2 1 2 77 1) c 0) c ) includes imaginary coefficients), is ( , supercurrent. They conjecture that , , such that any linear combination in y codes, and for i ) + w 1 )g( . Then the coefficient w ) understood as a map from codewords . When it holds the stabilizer group is c 2) 1 3 4.44 , understood as an abelian group under with all components being half-integer, ( c 30 = (1 / = (1 n − , c = 1 z 6 ) ) g( ∈ C 2 3.14 2 R (w + 1 c ( N and c 2 6 q ( ) ∈ + y k 0 , c c GF(4) ( out of – 42 – 1 1 , c goes to infinity. In the case of self-dual stabilizer k ~ c c z g( )) + n σ ) exists for any stabilizer code, but it depends on ... 24 c , c q )g( ) + w ( I i C ⊂ , 2 1 2) x c / 3.22 w ( , c → → y 1 . It should be real, which is a consistency condition on = 3 c ) − + 1 1) 0) c ( (w ( n ) , defined via an analog of ( , ,  1 y 2 C r k c w ( ψ ( . Choosing different combinations of the r i x ) = is in general nontrivial, but in the example considered in [ = (0 = (0 . This condition is symmetric under g 2 )+ → | c c . For 1 given by ( c c ( z k ~ ± . This is because P w )g( ) + w C . Here we point out this is not a unique situation, and in fact other 1 q 1 to generators is not a representation, but a projective representation c ψ c p, q, r ) = ( )+ . For the cocycle to vanish, n ) mod 2 c 2 ). g( ), and therefore the analog of ( x ( 2 with x c ) = 1 , c (w parametrized by vectors 2 w generators i 1 c 4.52 + p 0 2.66 p p, q, r | , c ( i 2 n SCFTs are related to other stabilizer codes. are integer numbers between 1 GF(4)  1 / P c c 3 = ( . These vertex operators can be associated with the ket vectors of the Hilbert ∈  2 =  ) = 1 = ( / C 1 p, q, r c N ψ ± α, β There is a particular technical aspect emphasized in [ = = ( i codes satisfy ( 5 Bounds, averaging overOne codes, of and the holography information central when questions the number of of qubits coding theory is how well one can protect (quantum) and should hold fora any genuine two representation codewords ofaddition. the In code general this conditionFocusing is on not real invariant under self-dual codecondition codes, equivalence transformations. for we have some verified that all codes with the code and where the group of Pauli matrices. Let us choose where equal to The cocycle vanishes, codes also have a vanishing cocycle, with an appropriate choice of the map from state the choice of result in a different c space of the Hexacode, the Hilbert space is mapped toshow a that linear the combination code of subspace vertexto operators. Hexacode Harvey and ( Moore mapped to the specialother vertex operator, the investigation. Here we onlyrelation briefly between the comment Hexacode, on understood as theSCFT. the recent quantum stabilizer work code, The [ and SCFT adimension particular in question, thek GTVW theory [ codes can be used to define non-chiral vertex operator algebras, a subject we leave for future JHEP03(2021)160 , ] by (5.3) (5.1) (5.2) → ∞ 59 numer- n ], where d 50 Even when 6 , . 4 ]. exist. 1893 . in the limit 41 0 → ∞ , invariant under the the results are shown We find that consider- n ≈ = 8 d/n for which the linear pro- 5 32 ∗ q d ], we find that the linear over the set of such poly- d , n ≤ d . In the latter case our linear 33 48 . , p n , 0 47 [ of degree ] was the additional condition that the . = 19 are possible [ . ln(3) ) 59 n d n ∗ q . For 30 p = 0 d/n ≤ − was considered. ] and [ x, y, z =0 n ( ) y

50 ⇒ ) ], which aims to establish universal bounds W x, y 92 ( mod 6 = 1 l – ) , y, y – 43 – W ) = ln(2) , n ∗ (1 78 ∂y ) readily provides an upper limit p , ( ( W 10 3.4 l – ∂ 8 1 5 (mod 6) saturating the bound exist, there are usually many other , while no self-dual codes with − =1 . ≡ l X d 8 d n d ,H δ ≤ ), with all coefficients being integer and non-negative, and d → ∞ 3.32 . We additionally want to maximize + 2 +  6 n , n ∗ q  p 0) = 1 2 . To illustrate the main idea of our approach we consider the following 2 , 0 ≤ 32 , ≤ , with at least one known exception when d d (1 d ≤ n ) except for certain values of . Comparing with known results for W n is the maximal achievable Hamming distance. 5.2 1 d The linear programming bound discussed above can be thought of as a toy version of Our first task in this section will be to obtain linear programming bounds on The quantum Hamming bound ( Self-dual codes are detectionOne codes, of so the they important are ingredients automatically in non-degenerate. the analysis of [ We thank E. Rains for a discussion on this point. 4 5 6 restricting our analysis to the subset of code theories defined in the previous section.coefficients Then of the so-calledsatisfied for shadow real enumerator self-dual be codes, integer and and therefore we positive. do not This discuss condition it is here. automatically “fake” refined enumerator polynomialsconstraints which and have satisfy the the same linear programing optimization the conformal modular bootstrapon [ the spectral gap and other similar properties of the 2d theories. Here we are essentially programming bound is mostly tight,gramming meaning problem that is the feasible maximal isthat also value achievable of by a codeprogramming (or bound potentially many gives codes) with extremal codes, i.e. codes with This is a slightthe modification feasibility of of enumerator the polynomials ing linear the programming feasibility bound offollows considered refined ( enumerators in somewhat [ strengthensin the figure bound, which mostly nomials, which can beproblem formulated as the linear programming optimization (or feasibility) ically, for problem: to findduality a transformation homogeneous polynomial ( satisfying analytically treating the linear programming constraints, Further improvements in the asymptotic bound for where but it is known to be conservative. A stronger upper bound was found by Rains in [ codes this is the question of determining the largest possible ratio JHEP03(2021)160 . y n (5.4) → − y , 30 B . n ) , but it is known to z n n 2 + ), as well as x -form codes, i.e. codes ( B 2 25 3.32 − n ) possible matrices z for the given 2 / d + 2 n 1) ). 2 y − 20 n 5.4 ( − n ]. It is an analytic constraint following from x 2 ( 59 + n ) – 44 – z 15 -form codes ( ) with all + 2 n B 2 y 4.26 + x ( ) found by Rains [ + 5.2 10 n x . Yellow: Gilbert-Varshamov lower bound obtained from the refined ) = = 19 it coincides with the actual maximal from below. Similarly to the classical case, to obtain the bound we n d 18 x, y, z 5 ( ≤ reduces it to the averaged enumerator polynomial, from which we immediately W n y asao bound Gilbert-Varshamov bound programming Linear ) eq.(5.2 bound Shadow = . Upper and lower bounds on the maximal Hamming distance for self-dual stabilizer codes. z . For 32 There is also a nonconstructive Gilbert-Varshamov bound for quantum codes, which Linear programming bounds are not constructive. In practice, they may be used to d is manifestly invariant under the duality transformation ( 4 2 8 6 ≤ 10 12 W Taking the code, which is exponentially difficult. bounds maximal calculate the refinedspecified enumerator by polynomial the generator averaged matrix over ( all trivial modular bootstrap analysisinvariant to polynomials. a simple linear programing problem inproduce the space invariant of polynomialsactual with code the associated with desired thispossible, properties, polynomial the is but only a universal verifying difficult way task. if to While make there various sure shortcuts is the are polynomial an is not “fake” is to construct n be conservative for enumerator polynomial averaged over all the partition function is fully specified by the refined enumerator, which reduces the non- Figure 1 Blue: “shadow” upper boundthe ( linear programming bound. Green: linear programming upper bound solved numerically for JHEP03(2021)160 - n ∈ is =  B n 2 2 ` (5.7) (5.6) (5.5) ∆ a, b R ], it is , ) ⊂ 17 , ) a, b C , where the 10 2 [ 2( for which the Λ( / 3 √ , which is twice b ∆ ∗ q d p = ≥ we find the maximal 2 ` n d/n ) saturates at one for . Even though the Gilbert- ), can be interpreted as the 1 5.6 via the Gray map. For large . C 4.23 . . for which the following constraint 1 , for which the coefficient in front ∆ their contribution to the density is d ∆ for which coefficients of the averaged < → ∞ . GV 1 . The binary Hamming distance of a b ∆ n d ) in figure )! − d c l n ( ) ! y ∆ 32 − n n n n Chern-Simons theory in AdS ) Γ( , n ≤ !( π ∗ n l increases such that the sphere of radius p n (2 ), there is no known systematic construction for 1 – 45 – ) + 2w d ∆ c = n is bounded from below by − ( U(1) become of order one, 2 for z l b n d ) 3 d ) 2 ∆ = × -code theories. In light of recent results relating the d n ), as expected. For a general 1 n ( B ≡ − , xy -dimensional sphere, yielding 2 =1 5.1 (∆) n l X GV ) := w GV 1) . When ). Similar lower bounds can in principle be obtained by ∆ assumes its asymptotic form. This might be different from for arbitrarily large 2 ρ c d d isodual code defined from , y ] we define the spectral gap as the value of U(1) n ( 2 4 − ] 2 b 3.3 x , the density of vectors of a unimodular lattice 83 b / d ( GV n n 1 (∆) C d ρ (2 the maximal . For vectors associated with codewords,  W ≥ 2 / n, n, d d ∆ 2 [2 ` → ∞ associated with a stabilizer code, all points of the form , until the overall coefficient in front of ( n ) n C 2 ∆ = / ) divided by Λ( 1 5.6 primaries. Following [ ) becomes of order one. This is the value of includes new codewords, the overall coefficient grows with each codeword eventually n The averaged refined enumerator polynomial, via ( belong to the lattice and for sufficiently large 5.6 2 to be equal to or larger than the maximal value / n . The spectral gap can be defined as the value of n of ( enumerator polynomial For a lattice Z given by ( ∆ contributing of a classical binary and sufficiently large given by the volume of a the dimension of theto vectors lightest in nontrivial the primary.norm-squared Lorentzian lattice, Primaries and of their Narainbinary dimension Hamming CFTs is correspond weight proportional is tocode the (minimal Euclidean weight of all non-trivial codewords) is the conventional Hamming distance average over Narain CFTsnatural to to ask if thea averaged weakly code coupled theory bulk may description haveU(1) is a possible, holographic we interpretation. would To like seedensity to if of calculate the primary spectral states gap of Varshamov bound is asymptotically weaker than thepresumably linear the programming actual upper maximal bound (and valueproducing of codes with averaged partition function of all This is a somewhat strongerwould bound than naively the follow conventional Gilbert-Varshamov from boundaveraging which over ( any classform of codes codes proves for to whichWe be this plot a averaging the convenient is numerical choice, feasible. for values which Averaging over an analytic answer can be found. smaller than the upperd bound ( is satisfied, conclude that for JHEP03(2021)160 ) 5.9 ], so (5.9) (5.8) , it is being by an (5.11) (5.10) 48 8 ) [ ) . Using τ τ ) ( ( /τ τ n 2 ( 1 2 n/ /η n/ 1 E ) as E → − 5.8 τ , for which is divisible by , and -form codes is a good n geometry, parametrized B = 24 , sp we expect the corre- = 1 3 n + 3 2 n . 6 τ ) a , + γτ → ) ( ) 6 τ 12 γτ E ] requires further checks and clar- 1 ( γτ ), which differs from 2 ( . n 6 ) η a 17 24 , ) , additional degrees of freedom in the τ ) η 2.29 ( τ + , a Z 2 , ( 10 24 ) n 12 τ n/ ), ( η ( a E SL(2 2 6 ) \ X Z E , generated by 2.51 ∞ 2 Γ 6 – 46 – ) = ) ) ∈ , which suggests the following picture. The class a τ Z γ τ SL(2 ( ( , \ X is exactly the Gilbert-Varshamov bound for binary associated with doubly-even self-dual binary codes. ∞ 24 ) + 2.1 ) Γ ) = η τ ) CFT n ∈ ( τ C γ Z ( gauge field, would be necessary to make the sum in ( (2 12 Chern-Simons on thermal AdS PSL(2 3 / Λ( = E b n Z of the boundary torus. This holographic interpretation is being unimportant. This expression can be represented in a CFT is not divisible by d 12 2 a Z τ 6 n codes a ) is modular invariant. Otherwise, since U(1) = Z , see section ] ) scales linearly with the central charge b 5.8 and ,( codes 5.7 and interpret this sum as a sum over handlebodies, with 12 Z ), 24 a 24 n, n, d in ( 5.9 [2 b d divisible by divisible by The calculation of the spectral gap above is somewhat schematic. What is important n n with the values of way similar to ( Their averaged theta-function is givenappropriate by modular ( form. For simplicity we consider the case schematic, and similarly toifications. the non-chiral Furthermore, if casebulk, [ possibly in the formwell of defined. a We overlook theseaveraging important specifically nuances, over code as CFTs ourconsider would goal change even here the is self-dual story. to In lattices understand the how chiral case, we would for the partition function of by the modular parameter For invariant under the subgroup of the conventional representation for the Eisenstein series we can rewrite ( The averaged theta-function is knownthat to the be averaged given partition by function the of Eisenstein the series corresponding chiral CFTs is for us is that sponding averaged theory to begravitational holographic. theory for We leave the the future,ries task and while of speculate here about understanding we their the consideraverage possible over dual the the gravity simpler Narain dual lattices case description. would of The be chiral chiral the average theo- analogue over of even the self-dual Euclidean lattices. self-dual codes of binary self-dual codesrepresentation, in obtained the statistical through sense, thewhich of are all Gray not self-dual map codes. self-dual from self-dual either Apparently, codes, the share or isodual a their codes overall similar number distribution is much of smaller Hamming than the distances number with of self-dual the codes.) (As a side note, the value of JHEP03(2021)160 ) Z , ) we , one , but gauge d , as is n . ). The n SL(2 5.10 satisfying 5.3 = 4 ]. 2 6.10 4 n (dashed line in ( we plot the U(1) 17 2 , b 6 2 4 E b 2 2 + 2 6 for a given value of a n/ b + d . can be written in many 12 E ) for the binary Hamming , and correspondingly the 3 12 , code theories fall short of 4 a = 4∆ = 24 5.2 E 3 b with arbitrary 4 n d = 12 a 2 4 n E + . 2 , for which there is a unique even 2 4 n 6 b E 2 + = 8 6 . For 4 a n . This is indeed the case for E 4 + 6.4 b 12 + = 12 6 E – 47 – n E 12 2 a 6 a and is not unique. Instead of + obtained by imposing constraints similar to ( 6 12 ], translated into units of a b = 4 . Superimposed with the approximate theoretical fit of the ,E d 10 8 4 n E ≤ there would be more representations and potentially a larger n can be interpreted as the holographic contribution of n ) τ ( and discussed in section dimensions and provide valuable insight about scaling for large ]. Similarly to classical binary codes, which give rise to optimal sphere 24 similarly admits a holographic interpretation. It therefore provides a 2 . Modulo the important subtlety that to make the sum over ) will become 8 24 10 /η ) – E 1 τ in the numerator suggests presence of non-abelian gauge fields in the bulk, 8 ( 5.11 8 6 . Thus the holographic partition function for and 3 . The question of finding a quantum code with maximal E /η 8 4 , in particular for n 8 a ), we find that both exhibit approximately linear growth with a similar slope. We n E = 2 behavior. In terms of real stabilizer codes, maximizing the spectral gap is equivalent 2 4 n b By comparison with the case of classical codes and Euclidean lattices, we may expect To conclude this section, we return back to the question of the spectral gap, which Even more intriguing is the chiral case of Our considerations do not prove that the averaged code theories have weakly coupled + is very similar to the conventional question of finding the “best” code with largest 4 quantum codes to yieldvalues Narain of CFTs withevident the from maximal figure spectralsaturating gap the for spectral certain gap special bound, which is discussed in more detail in section linear programming bound on bound is tight at leastnumerical for spectral gap bound [ in figure leave it as an opendistance, problem or to at find least the its analytic asymptotic analogue behavior of for ( large large- (up to an important nuancefor discussed a below) given to maximizing then binary Hamming distance to our knowledge it was not previously discussed in the literature. In figure lattice, and in thisof way lattice the sphere maximal value packingnumerically of in in the [ a spectral given gappackings dimension. in is related This to questionmay the expect has efficiency quantum been codes to recently occasionally studied saturate the spectral gap bounds and inform the setting to study fromwhen the the CFT boundary side is interpretation not of connected. the Euclideanwe wormhole geometries now defineequivalently strictly defined as as the the length-squared dimension of of the shortest the non-trivial lightest vector non-trivial of the primary. Narain It can be duality web in the bulk. self-dual lattice well-defined, additional degreesfunction of freedom in the bulk would be needed, the partition as a polynomial incan terms write of the mostnumerator general in ( expression b different ways, suggesting there areby different dualities. microscopic descriptions For in larger the bulk, related fields, while which are known to produce certain combinations of Jacobiholographic theta-functions duals, [ but they indicatethis that discussion such with an a interpretation few may concrete be questions. possible. First, We the end representation of a modular form Again, the term JHEP03(2021)160 . b d 12 n ≤ of real b n to CFTs d b we restrict d 30 = 3 n , but for larger 4 ≤ b . A similar strategy can ) d 25 1 − n 2 0 which are shorter. This is the , we focus on real codes, discuss ) 1 n 1 . Dashed blue line: an approximate ± n − n 2( = 2 2 20 0 √ , n 1 go over all classes of T-equivalent codes. ± 4 2( , we emphasize different points for different , n √ = 3 – 48 – 15 n , which precludes classical binary codes from yielding . ]. Yellow points: Gilbert-Varshamov lower bound on binary ). 2.1 10 [ 5.4 6.10 10 = 4∆ of the numerical bootstrap constraint for the maximal value of spectral b 2 d / in section 8+1 4 n/ 5 > we discuss all codes in detail. For b d ∆ = ierpormigbound programming Linear bound Gilbert-Varshamov = 1 . Blue points: linear programming upper bound on binary Hamming distance n b 0 In the discussion above we identified the spectral gap (length of the shortest vector) d 5 10 15 this problem can be partially solved by applying the shift procedure to the Construction . For In this section weAs discuss the many number explicit of examplesn codes of rapidly self-dual grows stabilizer with codesthem for in detail andour then attention illustrate to the B-form coset construction. codes, and Starting for from be employed in the quantum case,with making larger it spectral possible gap. to relatecontrolled We codes by consider with an larger explicit example of a lattice with shortest6 vector Enumeration of self-dual codes with small there are always latticesame vectors problem, of discussed the in section form efficient sphere packings in large dimensions,n at least directly via ConstructionA A. lattice, For small which will remove unwanted short vectors gap, measured in unitsHamming of distance obtained from ( with the binary Hamming distance of a code. This is correct for Figure 2 self-dual codes. Ittheoretical is fit tight for small but presumably not all JHEP03(2021)160 5 ), ≥ (6.5) (6.3) (6.4) (6.1) (6.2) . The . The . The n 3.36 1 = 1 = 1 n n , we give explicit . The interlace 4 ! we give examples 8 codes, with stabi- / , ) includes two ele- , see figure 12 1 1 1 0 , = 1 = 7

6 4.36 = 1 , 4 = 1 n n / d , is T-dual to the first one. b = 4 d ! n . For b 0 1 1 0 ) works in the case . Its lattice generator matrix is d g = Z ∆ =

. . ), = 2 4.36 . The coset ( ! 2 ! , ) . R Z ! , . 0 1 1 0 and/or 0 1 1 0 (or 1 y , in agreement with the spectral gap of z. or Z ,

d

0 1 1 1 , ) is equally straightforward. The group + + (1 ) correspond to two

± 2 = 1 x , = 1 x O . This is the only B-form code for b 4.37 ! = or Y , , – 49 – = R d ), specified by the unique stabilizer generator 4.33 , 1 1 ! ! 1 0 0 1 Q . The third code W 3.33

is B = 0 1 1 0 1 1 0 0 1 g = X = 1 =

correspondingly. R ± F , B = 0 ) with ! g = Z ) are equivalent, they share the same enumerator polynomial 1 0 1 1 4.28

6.1 and , . The refined enumerator polynomial of these two codes is, cf. ( ! 2 . The Hamming distance of all three codes is consists of four matrices g = X y ), the first and third, are real and correspond to code CFTs. The first code √ ) 1 0 0 1 / we also explicitly write down all “fake” enumerators. Starting from Z +

, 2) 4 x 1 3.35 includes six elements, , there are three codes, see ( , , ) 2 = 1 is a B-form code ( Z ) = = 3 O(1 = 1 n , 1 n The coset description of all codes ( We would like to see how the coset description ( n , x, y ( = diag(1 lizer generators O(1 The resulting lattice generator matrices ( The first two matricesments, form with the representatives subgroup binary Hamming distance ofthe compact real scalar codes onpolynomial a of the circle graph of with radius group Λ Since all threeW codes ( codes, see ( g = X corresponding graph is simplyis a a graph boson consisting on ofThis a one is circle vertex. a of boson The radius corresponding compactified CFT on a circle of radius For Obviously all three codes are equivalent (in the sense of the code equivalence group). Two we only consider codesexamples with of maximal non-equivalent codes values with of theto same refined groups enumerator of polynomials, giving physically rise of distinct codes isospectral related code to CFTs. special For lattices. 6.1 For JHEP03(2021)160 . . , 2 2 B z = 1 or (6.7) (6.8) (6.9) (6.6) R ) is of + 2 right, 2 3 = 1 6.8 y )). They matrix R + 2 6.4 2 ) (left) and the × x ) is T-dual to 2 . Therefore the , 6.7 = 6.8 2 2 √ ! I Z) W , 0 1 1 0 = (Z I

W and the (binary) Hamming 2 , 1 B = ), are real. We only list real . The first code in ( W I X) = 2 has the toroidal compactification 3.35 = , . ) b 6 . d C W ) as follows (Z I ! ↔ . Multiplication by the only non-trivial = Λ( 1 ) 5 6.5 ) is of B-form, with zero . 2 d . The second code in ( − elements. All codes within each group are ) Z , 2 y , , 6.7 1 1 1 1 4 left. The interlace polynomial of this graph Z Z) , +

I Z) , 3 , ↔ and (1 x = – 50 – ( 2 3 4 x γ O (X I (X X , = = 2 2 , ∗ . Six of them, see ( . The corresponding graph is shown in figure γ 2 , ↔ 2 ij Q  = 2 1 Z X) I X) , = n , -field and is the square lattice with minimal length ! B ij γ . B 0 0 0 0 Z permutes the matrices in ( ,

. Only the first code in ( ) ) = (X I ) = (X Z Y . 2 2 2 , 2 -flux, g g ) Z = 1 , , , y B B = 1 1 1 b , + d (g (g ) (right). g = X x (1 ), and (binary) Hamming distance = 2 6.8 d O = ( ) with vanishing 3.36 2 1 . Graphs and their adjacency matrix associated with the first code in ( = 2 Q 4.11 n = Codes in the second group are indecomposable. They have Q The lattice generated by corresponding CFT is the tensor product of two theories, each being a boson compactified cf. with ( B-form. The corresponding CFT consistswith of one two compact unit bosons of and on circles the of interlace radius polynomialthe is first one. Theform ( generating matrix of its lattice The refined enumerator polynomial ofdistance these is codes is The corresponding graph isis shown in figure The codes are split into twoT-dual groups, to with each other.code The CFTs first are group tensor consists products of of decomposable two bosons codes. compactified The on corresponding circles or radius 6.2 There are fifteen codescodes, with which are specified by a pair of stabilizer generators, element of Therefore matrices 1, 3,correspond to and 5 can be chosen as class representatives (cf. ( Figure 3 first code in ( with the first two forming the subgroup JHEP03(2021)160 2 2 as S × γ × , the 2 (6.15) (6.12) (6.13) (6.14) (6.10) (6.11) 2 1 i S , τ ), we find =¯ 0 ) reduces to ) with τ

. 2.59 2 , 0 2 is arbitrary. In ! c 6.10 4.11 ) of real self-dual Z 2 2 c are related to each S 1 0 + 0 1 4.36 − H . The theta function (these are matrices of 2 ) 0 4

) b and 2 2 2 ∈ b = S , xy 2 2 i + × , y 2 , 1 factors mod 0 2 2 ,  S a ) , x ) generated by ( , ( 2 ( ) t R a R 2 2 . Comparing with ( , Z , , = 1 , 2 , which is the correct theta function 2 W mod 2 = 0 = , τ R 2.2 1 . 2 − 2 c , as expected. SL(2 ) and ! 2 1 , i ) = SL(2       √ ) = φ SO(2 2 / 0 × q τ = 1 x, y associated with (¯ ∈ ! ( , the Siegel theta function ( + 3 4 i . 2 4 3 where τ (2) c θ i ! 2 0 2 det = 1 1 i ). This is a curious situation because the code ) 0 3 R / √ b φ 1 2 2 W − − – 51 – q − S 0 0 10 0 1 01 0 0 00 0 0 0 1 − =  ( − ) 4.23 2 4 i = 3 2 T 1 2 ×       θ S ) = Γ d ) is 0 4 = 1 S C i 1 − . This is confirmed by the partition function 1 √ τ Z ( 2 / b = 2 S 1 , b

) + t after setting 2 d √ d 2.22 2 , . 4 , a , we describe the coset construction ( q 4 b 2 (¯ = mentioned in section (2 3 ! 2 R 2 θ i i is the product of two H S S R = 2 ∆ = ) O b 1 d 4 1 in ) 2 c n a i i q R 2 ( c a − includes only matrices with all elements of the upper-right , 3 √ 2 ) θ

. The Lorentzian lattice in , ) can be described in a similar way. They include products Z

(these are matrices of , = = ) 2 ) i = Z ), reduces to 2 , Z , S SO(2 2 S , 2 (2 x, y, y √ , 2 ( plane. Upon setting × 6.10 2 = O 1 2 R C W SL(2 S O(2 Z = 2 ∈ | 2 ) 2 ), where , τ ) = 1 1 R ( S ), and the Euclidean lattice in − η | x, y To conclude the case of Using the Gray map, we can also interpret this code as a binary repetition code, such This code has other interesting properties. It is in fact the repetition code ( 6.9 2 i The subgroup submatrix being even.other This words leaves Elements of where det = codes. The group the theta function of the Euclidean lattice that the enumerator polynomialfollowing ( from ( of a cubic lattice of size other by a linear transformation in each spectral gap of this theory is linear self-dual code ofW type in ( where the right-hand-side follows fromCFT ( is a tensor product of two theories, while the code itself is indecomposable. The on a circle of self-dual radius JHEP03(2021)160 , ) γ (2) 0 , the 6.19 Γ (6.19) ( (6.20) (6.21) (6.16) (6.17) (6.18) 1 / . Both , ) 3 γ Z y , = 0 + 4 l 2 , SL(2 3 ), with some z xy 4.11 + 4 + 3 2 3 -field, but it is almost , . When I x are T-dual to the code . orbits under T-duality B xy ! ), with the same × = 3 3 ) = + 3 = 2 n 1 0 = 4 l 3 . The classes of T-equivalent 4.11 ), are 0 1 ( x − . The quotient ) 6.8

= = 3 2 x, y, y C , s ELC 4 correspondingly. ( 3 but different binary Hamming dis- t n 3 = I) ), ( Λ( 4 ˜ ˜ W 3 / W × . 6.7 = 2 3 . s    d , s ( ) =    ∆ = 3 ! 1 0 and and some non-trivial − . 1 1 0 1 x, y, y 0 1 1 1 0 1 1 1 0 and γ 0 00 0 00 1 ( ,. t -field and are isomorphic for – 52 –

3 = I 2    I) I)    l B , / W = γ 3 = × × = z 2 1 2 γ , which split into s s + B ( ( 2 ∆ = 1 , s = 3 ) with xz ! by changing n 4.28 , t , t + 3 0 1 1 0 B are related to each other via C-equivalence. Accordingly, their I I z

2 × × . The corresponding CFTs will have different partition functions, are T-dual to the code lattice . The equivalence of codes within each orbit is obvious, as the y 1 2 3 = inequivalent classes of codes for = 3 -field ), with vanishing s s l 1 = 3 B + 3 ≤ s = 3 b 3 l l = 3 4.11 d x ( and -field. This is the general situation: it is always possible to use T-duality ) ≤ = LC 4 t C B real codes for and 3 0 ) to the form ( is the Hecke congruence subgroup of level = 2 Λ( W l 30 = 2 4.11 (2) = 3 0 b d Γ n There are yield the same enumeratorclasses polynomial of codes have thetances same Hamming distance as well as different spectral gaps, never possible to get rid of codes with refined enumerator polynomials In this latter caseand there vanishing is no T-dualto theory bring with ( lattice of the form ( B-form codes with and with non-trivial as their properties werewith discussed lattice above. B-form codes with There are equivalences (T-equivalences). Incorrespond terms to of the the fouredges B-field inequivalent (links) graphs representations, with these threegraphs four vertices, with orbits labeled the by samegraphs, the and number number hence of of the links codes, are decomposable. We will not discuss these cases in detail 6.3 includes three matrices, Six real self-dual codes, arranged the same way as ( where JHEP03(2021)160 , y ]. An ), and (6.22) (6.23) (6.24) (6.25) (6.26) (6.27) → −    96 y – 3.36 1 , there are 93 Q 2 4 0 1 1 1 0 1 1 1 0 , unless they 3 x n ˜    W ) and = = 4 , besides the four 3.32 B = Q W = 3 n are defined in ( 3 1 , , Q ,W 3 ) 3 , . , , 2    3 z 3 y z 3 3 z z z z + ,W + 2 1 + 2 + + x + 3 + 3 2 z z z 0 1 1 1 0 0 1 0 0 3 W 2 invariant under ( 2 (3 2 2 z 2 xz y y y    W x , all coefficients integer and positive, ) xz y + = = + 2 + 2 + + + 2 2 2 2 2 2 2 B = Q W x, y, z xz 0) = 1 ( xz xz xz xy xy , 0 W + + 3 + . , – 53 – 3 + 2 + 3 + 2 1 2 z z (1 2 2 2 z z    W 2 2 x x xy W x xy x 2 + 2 2 + + + 2 + 2 + + 2 W Q 0 1 0 1 0 0 0 0 0 1 W 1 3 3 3 3 3 3 1 x x x x x x    W Q codes. The polynomials W ======+ 3 B = Q = 3 W W W W W W 3 W , all polynomials n W − = 2 = n 3    z + 4 and 2 3 0 0 0 0 0 0 0 0 0 1 3 1 xy    W Q = 1 . Graphs, their adjacency matrices, refined and interlace polynomials, associated with ) is modular invariant and satisfies other basic properties expected of the CFT n + 3 = = 3 x B = Q 4.23 W For ≡ 3 ˜ of these “fake” polynomialsrefined enumerator correspond polynomials to for actualcall non-additive CFTs, codes. code they In theories likely this shouldContinuing can case this be the logic be extended scope further, identified to the ofcode as CFT what include enumerator partition polynomial, we CFTs and function based may is satisfy on a additional much a non-trivial richer conditions wider object [ class than the of codes. partition function. Thisbootstrap poses program: the do following thoseories? question would-be Given CFT important that partition in the functionscorrespond light number correspond of to of to actual the “fake” actual polynomials CFTs, the- modular consistency “bootstrapping” increases regions rapidly 2d (occasionally taking with theories the mustare form yield of in “islands”) a fact in growing empty, the contradicting number exclusion our plots, of which experience so far. Assuming the opposite, that some There are no additivepolynomials, self-dual yet codes they for satisfy which allvia these necessary ( properties, polynomials and are the refined “partition enumerator function” defined and satisfying additional conditions, are actual refined enumeratorpolynomials polynomials associated of with additive the codes.another four six For classes “fake” of polynomials, T-equivalent codes, see figure Figure 4 the four T-dual classesW of JHEP03(2021)160 + 2 = 2 z b 2 (6.29) (6.28) d x = + 2 d y 2 x + , and the interlace 4 4 1 of which correspond x W matrices (graphs) = = 4 , , B R −             W 1 ELC 4 i W 2 − , and the interlace polynomial is 2 4 0 0 10 1 0 11 1 1 01 1 1 1 0 1 0 1 01 0 0 10 0 1 00 1 0 1 0 W       W 2       1 W B = B = R − 1 = 2 4 W z – 54 – − + , 2 3 , .       W 2       xz 2 1 y 2 W x + 4 2 ]. The REP for this class is z + 3 = 2 2 21 0 0 00 1 0 10 1 1 01 1 1 1 0 y y 0 1 01 1 0 10 0 1 01 1 0 1 0 4 3       z x       . matrices (graphs) + 4 2 . 2 / 2 B . Both classes of code theories have the same value of + 2 + 8 z B = y 0 2 4 2 B = 2 x z x x classes of T-equivalent codes in this case, 2 y ∆ = 1 + 2 = 5 + 2 R → = 9 z y + 3 2 Q 3 3 x xy ELC 4 ’s for the same class of T-equivalent codes to emphasize that edge local com- t xz B + 8 = 4 + 4 4 4 + 4 x n x z The two classes of T-equivalent codes described above are related to each other via The second equivalence class include B-form codes with the 2 = = 6 xy Q C-equivalence. One can easilysame check by that taking the enumeratorand polynomial spectral in gap both cases is the and those isomorphic todiscussed them. as an The4 example edge in local [ complementarity of these two graphs is polynomial is and those isomorphic (permutationdifferent equivalent) to them.plementation can We change explicitly theW write number down of two links, topology, etc. The REP for this class is to indecomposable codes. We discuss onlyB-form the codes indecomposable with ones. the The first class includes “fake” code enumerators from actualcode ones, theory. thus introducing a new string theoretic6.4 tool to There are interesting scenario would be if these additional conditions could be used to distinguish JHEP03(2021)160 , ) = C 4 = 2 z Λ( n (6.30) (6.31) +2 theories and is a 2 , and also 4 z associated . In other 2 2 8 but usually ) y ij = 3 E δ d as the length- U(1) − n +6 4 and the interlace × x, y, y y ( . = 1 4 2 3 + R 2 = 2 2 ij 1 W / z xy . B b 2 W U(1) d x + . B-form codes from the ) = 2 + 4 y +6 2 2 4 2 ∆ = 1 x, y x = 3 x ( W ). Accordingly, the enumerator n = W + 4 = y = 2 4 W 3 z x LC 4 i + 8 + 7 . Codes from the third and fourth classes is in fact the optimal (maximal possible) 4 . The corresponding Narain CFT, which 4 y x 6.7 + = 2 binary code, and the Lorentzian lattice 2 = 4 2 y ` – 55 – ∆ = 1 2 Q 4] ) we immediately recognize that the binary code . Now we can recognize x ) , 4 . , , 2.18 theory, has spectral gap + 6 x, y [8             ( lattice 4 8 8 e x 8 E E W = . 8 ) with ( ) qe y ) = correspondingly. This is similar to the case of 1 0 11 1 1 01 1 1 1 0 0 1 1 1 1 0 01 0 0 01 0 0 0 0 0 1 1 1 W + 4 6.31             , xy x 2 ( 3 , y and x B = B = 2 from ( 2 x ( , . = 8 8 B b b ] as the CFT saturating the bound. Here we have found that the non-chiral qe d d Q and interlace polynomial is 89 W R is an even self-dual lattice in both Lorentzian and Euclidean signatures! As a 1 ]. The theory of eight free Majorana fermions with diagonal GSO projection was 8 W E 10 2 Modular bootstrap studies of the maximal spectral gap in Comparing The third and fourth classes are related to each other via C-equivalence (thus there Finally, the fourth class has a unique B-form representative with There are two more classes of T-equivalent codes for − 2 2 we will refer to as the non-chiral reveal with an astonishingvalue precision [ that identified in [ dual. It isunderstood the as extended a Hamming latticewords in the Euclideanconsistency space, check is one the can evenbinary self-dual easily code lattice verify thatsquared the of enumerator the polynomial shortest of root the of resulting have different reflection of the generalhave situation: different C-equivalent codes must have the same obtained from this stabilizer code via Gray map is not merely isodual, but in fact self- polynomials of the third andthe fourth same classes as are the the enumeratorcodes. same, polynomial This of is the an decomposableand example code different codes of consisting may a of share two genericthe the refined situation: enumerator same polynomial, enumerator enumerator see polynomials polynomial. section are The not same unique is also true for The REP of thispolynomial class is is are two classes of indecomposable equivalent codes and those isomorphic to them.W The REP of this class is with the complete graph with four vertices third class are given by JHEP03(2021)160 d and ) we (6.33) (6.34) (6.35) (6.36) (6.37) (6.32) WZW lattice, 1 4.23 8 ]. In fact 1 + 3 E 89 \ and ( SO(8) . 8 A qe W , for the decomposable ) to describe all of them = 4 , n = 10 3 , 2 , , ELC 5 1 ) for 8 , | n , i 1 c | , . − 8 1 1 + | ) = 21 = 8 2 2 W W | τ theory in the appendix ( b R R | 8 η | ELC 5 E + ± ± 2 t , 8 , 4 4 1 1 | 4 , k 1 R 2 2 a 1 | – 56 – W W R W 1 , ]. While the description of this theory as a Narain W + − − ) = 2 2 97 ± 1 1 2 k W ¯ = 4 τ 2 is a characteristic of the graph equivalence class under W W W n τ, + . 2 2 2 ) W 1 ( 2 2 2 x 1 2 8 ) W W ( W y E 2 3 2 1 W W Q Z + = = 2 = = = x ( 2 ]). We establish an explicit relation between the theory of eight x 8 W W W W W . Edge local complementation acts on each disconnected subgraph 5 = 4 Q “fake” REPs for 11 there are too many classes of codes ( . Two non-equivalent graphs under edge local complementation, which have the same = 5 n = 5 can not be equivalent. . This happens for the first time (meaning smallest The interlace polynomial There are ) n Narain CFT also saturates the bound. Using the explicit form of lattice was noted in [ x ( 8 8 6.5 For in detail. From now(the on conventional we measure will of quality only for focus quantum on codes) codes or maximizing the the binary Hamming Hamming distance distance edge local complementation (and isomorphisms),Q but different classes maygraphs share shown the in same figure individually, and therefore these2 + graphs, 2 which have different decompositions CFT has been discussedunderstood previously, as to a our Narain knowledge lattice,E the has connection not with been pointed the Majorana out (a fermions connection and with the the non-chiral Euclidean Narain which coincides with the partitionthese function of are eight the fermions mentioned samemodel. above theory, [ The which theory haswhich of other has been eight descriptions recently fermions including discussed exhibits in as [ a the group of symmetries, known as triality, E readily find the partition function of this theory Figure 5 interlace polynomial JHEP03(2021)160 , . ], ), = 39 = 4 Q = 5 ), the 2.66 b ) upon n d and 2.66 2.60 5 ). z code with the + ). For = 2 b classes of codes 3 d d , which also had z 4 2 b = 5 y d n . and maximal ) class of codes with the (and and + 5 y ), the so-called shorter 4 3 + = 4 , = 5 = 4 = 3 x xz ( 2 d b b 4 ]. Those codes are not real.) n d d x + 5 and so on. , but smaller , which reduces to ( 76 2 , 6 d z z 2 = 16 20 (and = 8 xy Q + 2 n . 4 = 3 z 7 and + 10 . As can be easily seen from ( 2 to the shorter Coxeter-Todd lattice [ d 5 y 3 z z 2 2.2 2.2 x + 15 , figure + 16 6 4 . The number of “fake” REPs increases rapidly – 57 – + 5 y = 4 z xy , 4012529 for 2 + , which are associated with the class of T-equivalent b = 5 y 4 d 2 8 + 5 z n x = 7 2 2 there are already groups of y x n 3 + 5 x 5 for = 5 x + 15 . And there is a unique graph (up to isomorphism), which 2 + 10 n = z we saw examples where C-equivalence would relate two classes 5 6 2 x y , introduced in section 71164 W 2 = 6 , x = 4 , h .
W n 2 “fake” REPs for , = 6 . y = 3 + 30 y n 6 b = 2 and + 5 128 d x → d for , z , xy (note, there were other codes with the same ) = = 3 d = 4 = 3 . ELC-equivalent (T-dual equivalent) graphs corresponding to . Unique (up to isomorphisms) graph corresponding to n = 6 + 16 2835 b d d 2 , , there was a unique class of T-equivalent codes with maximal n x n x, y, z 4 There are There are three distinct graphs (up to isomorphisms), which correspond to the shorter For ( 11 (which determines, up to some nuances, the spectral gap in the code CFT). For all 6 ≤ h 3 b which are also called byThere this are name two in graphs the shown literature, incodes see figure that [ includes theW hexacode. The refinedsubstituting enumerator polynomial of the hexacode is 6.6 There is a unique classThis of is codes the which hexacode achieveshexacode both is maximal a realB-form. code, (We and should by note using that T-duality there transformations are it other can codes, be C-equivalent brought to to the hexacode the ( of T-equivalent codes. For related to each other by C-equivalences. with and there is another unique class of codes withhexacode the class, maximal seecorresponds figure to the second class with d n maximal there is a unique classhexacode of related codes via with Construction A the of maximal section Figure 7 largest largest x Figure 6 JHEP03(2021)160 , 3 ). y , 4 2 x 1 2.67 ( , W +2 ]. That 4 2 1 2 2 12 y 10 W 2 W 5 K x , is not the R R graphs (up to 12 − − 3 +24 K 4 1 1 y , which we do not 6 W W x 2 4 2 R plane. Upon setting = 2 R 5 +62 d . Let us first focus on 7 C − − x 7 7 1 1 . = 4 and 4 2 4 W W y = 40 b classes attaining the maximal 5 5 5 d x Q = 4 − − 18 b 2 2 2 2 d + 21 2 2 W W y as a Lorentzian even self-dual lattice, 6 R , although related to R x 8 ) . The first class includes 6 is related to the Euclidean Coxeter-Todd − E − b h . ) d ) applied in each 2 1 3 6 1 1 + 64 . We only show one representative for each Λ( reduces to the theta function of h 7 6 W W W ) x 2 2 2 2 4 6.11 4 2 Λ( and 6 R – 58 – h W d W 7 11 R = 43 Λ( . The REPs for these codes are (from left to right) Q + + − 9 7 1 1 1 4 W W W 2 2 3 3 , and 3 2 2 3 . and the third ) y W W − 2 4 y x 2 12 1 + + + 3 3 W 1 1 + 5 2 2 classes with the maximal value of y 2 . W W 4 4 5 classes of T-equivalent codes, with xy W 2 2 2 2 8 x R W W 6.4 7 + 16 + 23 + + 2 − = 218 y x . The interlace polynomials for these classes are 5 5 , and 1 1 1 6 x W W W (11 by the linear transformation ( = 4 2 2 2 2 2 4 4 ELC 7 = 3 t b x W R W , the Siegel theta-function of d 12 + 63 . ELC-equivalent (T-equivalent) graphs corresponding to the hexacode. These graphs d . Representatives from three distinct classes of T-equivalent codes, which maximize both 3 7 3 = 7 τ classes which have both maximal 7 K = 2 3 x n 3 = = = and − Q There are two other classes of codes with maximal The Lorentzian lattice of the hexacode = W W W = 41 = 3 0 value of those isomorphisms), the second class of T-equivalent codes in figure discuss here. 6.7 There are same as the Coxeter-Toddlatter lattice interpretation understood and as the aconstruction related Lorentzian is Narain even analogous self-dual to CFTdiscussed our lattice. was in construction recently section The of introduced in [ lattice τ We should note that the Lorentzian lattice Figure 9 d Q Figure 8 have JHEP03(2021)160 . 7 . In = 7 z 16 (6.38) n R + 2 3 6 z . We choose 4 xz 9 x + 4 + 10 2 5 z z 4 2 y xy . 3 + 9 + 11 y 4 2 5 x z z 2 2 y x + 6 3 even self-dual lattices in 2 x y + 16 5 ]. Our construction gives an exam- + 11 x D 4 + 12 31 z – 2 2 z + 34 and 29 5 xy y 8 x 6 x E + + 25 ⊕ z . Because they share the same REP, the corre- 4 4 8 – 59 – + 58 y z E 7 2 3 D ); they are C-equivalent but not T-dual to each other. x x x 6.38 + 5 + 5 = 30 z 3 2 z Q 4 y 4 y x + 4 . It is interesting to compare this example with the examples + 5 7 3 .) But it is also different from Milnor’s example in several ways. , distinct non-isomorphic graphs; the second class includes 2 z 7 7 2 y R 5 y 10 ≤ 2 x x n + . (We note that the REP is unique for all classes of T-equivalent codes 7 for +22 x 10 = n,n R . “Fish” graphs — representatives from two classes of T-dual codes, which share the ). It should be noted that the shown graphs are related by LC, which means 6 ≤ Since these two classes of codes are not T-equivalent, corresponding code theories are It turns out, there are exactly two distinct classes of T-equivalent codes for isospectral n W First, Milnor’s example relatedwe a have decomposable two indecomposable code codes.isospectral with non-chiral an Second, CFTs, indecomposable at not the one.there related level in is Here of any no CFTs simple obvious itdoubts way symmetry is that to which an these chiral would example isospectral CFTs. of make theories Furthermore, might two this be example related unique by or a special, duality. raising ifications of 16 left-movingare modes related of by the T-dualityple upon heterotic of compactification string. isospectral [ even These self-dualdimension two lattices it isospectral is in theories unique, the much smallestcode-related like number lattices. the of Milnor’s dimensions, example. It andlattices (We is in in note that an our open analysis covers question only if there are other even self-dual isospectral trum. At the level ofLorentzian lattices, lattices this in is aof pair isospectral of isospectral but but not notcodes, isomorphic isomorphic in Euclidean even particular self-dual lattices Milnor’s associated exampleString of with Theory this inequivalent classical example famously corresponds to the two possible isospectral compact- C-equivalence can change the REP, in this case itnot does T-dual not. to eachsponding other, code see CFTs appendix havean the explicit same partition example function. of two In Narain other CFTs, words not we related have obtained by T-duality, with the same spec- Two representative graphs (thereshown are in many figure others) associatedfor with these twothat classes are the corresponding classes of codes are C-equivalent. And while in the general case which share the same REP, same refined enumerator polynomialThe ( first class includes among them two representativesalso based share on the simplicity interlace and polynomial aesthetics. These two classes of graphs Figure 10 JHEP03(2021)160 . . 4 B ) = = 4 . (6.41) (6.39) (6.40) 8 b z , xy 2 , d . , y d 2 + 32 -matrix (one x = 2 . 4 ( 8 B z 2 d z ) 4 8 y qe ( is understood as a and + 16 W 8 + 24 4 8 E z 4 2 qe 4 z is the code with lattice y 2 . This means the binary W . Fourteen classes achieve y 2 x = 0 = 4 + 28 = 8 8 ii b n y d B but they have smaller ]). The latter can be brought to + 144 + = 4 27 4 ). Upon bringing it to B-form, its z z d = 4 4 2 x in [ b y . They form 5 groups of C-equivalent 2 d 3.17 x 16 but this time each = 4 + 24 E 8 8 b ). As can be seen from its graph, this code is E y d + 168 – 60 – ⊕ + 4 z 8 6.31 6 4 y , E x 2 , and (denoted x                 . The REP of this code is matrix is block-diagonal, with each block equal to , + 16 + 28 = 4 + 4 d 6.4 ! B 4 4 d 4 y y 0 . As we have mentioned several times already, the binary 4 4 B 2 and REP x x ) is given by ( 4 -matrix being symmetric and 0 4 B ). Its B x, y = 4 B + 14

classes of T-equivalent codes with ( + 22 b 8 8 2 e 6.31 d x y , 6 W B = can be uplifted to the real self-dual stabilizer code with = x 1 1 01 0 1 0 1 01 0 0 1 0 1 0 01 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 11 1 1 1 0 11 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 of that code is also , understood with the metric ( 2 matrix = 2 8 ) ) + 16 = 1068                 8 + 4 ) = E 4 d d C codes ( qe 8 ( × ⊕ x Λ( ELC 8 , xy B = W 4 t 8 2 = = 4 ]. There are other classes with maximal E = 8 code is isospectral with , y 16 + n 2 75 d 8 . Our finding highlights a limitation of the modular bootstrap approach, which is n x e 8 ( W Among the fourteen classes with the maximal As we will see shortly there are many more examples of isospectral Narain CFTs with 8 ) = ⊕ C 2 ≥ qe -matrix is given by 8 This code has canonical form with the self-dual code of many representatives from the T-equivalence class) and graph The lattice Lorentzian lattice, as inBoth section codes would beW equivalent as binarye codes, and in particular indecomposable and its REP is We should note rightof away that two there is another decomposable code, which is a product where the the maximal allowed valuecodes of [ Λ( B n incapable of differentiating isospectral theories. 6.8 There are JHEP03(2021)160 + 4 z code 2 (6.42) x (purely = 2 τ + 4 2 − n z 2 = y 2 code associated ¯ τ x + 2 = 8 4 n y 2 x code with the isospectral + z 2 real self-dual stabilizer codes = 1 y 3 n x = 8 ) theories, and the one associated n + 4 2 z 4 pairs two relate a decomposable code 6.31 x 60 relate two indecomposable codes. Among , combined together with the + 3 that are not orthogonal to other codewords z . Among 8 58 5 8 . , x ) 8 2 6 = 2 . R z – 61 – b − d 6 11 , which means that all three code CFTs — the one + 2 x , 2 ( 4 ) 2 z                 ) 2 z y x, y ( + 8 ). Among these x e + 10 W 2 10 = ( z 4 y W ) = + 9 , xy ), the tensor product of two ( 5 2 xz , y 1 0 11 1 1 1 0 11 1 0 1 1 0 1 01 0 1 1 0 0 1 11 0 0 1 0 1 0 00 1 0 0 0 0 1 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 6.39 2 right, which is isospectral with the indecomposable x + 4                 ( 3 3 + 16 z d 2 ), but are different otherwise. B = τ does not lead to isospectral Lorentzian lattices. This does not mean there is any ) — have partition functions which coincide along the diagonal W . Representatives from three different not C-equivalent classes of T-equivalent codes, xy 16 codes shown in figure 6.41 R + 24 z We will not discuss other examples of isospectral pairs in detail, but just mention We just saw that Milnor’s example of isospectral even self-dual lattices in Euclidean 4 = 7 xy that in 36not. instances Besides isospectral 60 codes isospectralCFTs pairs, are are there C-equivalent, isospectral. are while 5 FourC-equivalent. isospectral in triples All triples, three include 24 codes when two in instances threefrom the C-equivalent different fifth they the codes, code triple fifth and are are triple not another C-equivalent. are one, Representative shown graphs in not figure One can easily find twoin codewords with terms of theinto Euclidean a metric. sum of Thisexample two means is lattices, corresponding similar but lattice to is Milnor’s is isospectral example. not with decomposable a decomposable one. In this sense this with the graph (one of many representatives from the T-equivalence class) there are 60 isospectral pairsn (excluding the product of the with an indecomposable one, whilethe the first other two casesshown is in figure the hexacode, see figure associated with ( with ( imaginary space lack of isospectral even self-dual lattices in Figure 11 which share the same REP 2 Of course JHEP03(2021)160 7 12 = 12 y (6.45) (6.43) (6.44) . That . The n 4 + B / b ]. Second, 10 4 , will make d z z 10 2 4 [ x y = 4 2 codes available 4 2 x ∆ = < ` + 60 ∆ , and one can easily 8 = 11 z -tuples of isospectral E 2 + 480 n k y 2 2 z is one of the codewords, x 6 y 1 ~ 4 x will yield a new even self-dual + 810 6 . As we have mentioned already, and 45144 + 60 2) z 2 codes, and 4 ], with the corresponding database / 6 √ y z 2 75 6 (2 x = 10 x / , seems to suggest the celebrated Leech 1 = 10 ~ 4 = 3 n 2 / n = b + 64 . d vertexes, see appendix + 840 . Nevertheless in certain cases one can apply δ 4 4 12 1 8 with no vectors shorter than z z z 2 , 3990 6 ≤ y – 62 – ≤ ∆ = y 6 2 obtained in [ + 4 n x ∆ = 9 = 24 x ]. Real codes belonging to this class split into three 10 n d 12 z 73 2 ≤ to attain larger spectral gaps. To turn an even lattice y for the future. There is a full classification of LC classes + 270 of any stabilizer code necessarily has vectors of length + 270 , a twist by 2 n δ ) n 8 z C z 4 + 60 4 y , saturating the bound. But this is not the case. First, the x Λ( 6 8 = 12 . z x 4 n y , , 11 should be odd. Assuming the vector = 12 3 3 2 + 105 ≤ + 60 R R n δ 6 2 2 6 1 1 k z y + 105 2 6 W W y 6 x 2 4 . Going through 675 z 11 6 x − + 4 y , depicted as a dashed line in figure + 2 − 8 . III III / 12 is odd e.g. for 12 W W x +840 +64 = 12 y online . It corresponds to spectral gap 4 = 9 ). This procedure is universal, and can be applied to any code whose REP includes + 4) codes for all = = = n , which limits the spectral gap to n/ n I n , the so-called the dodecacode [ = 6 II ( III Strictly speaking we should say at least three, as potentially there could be isospectral classes of There is a unique equivalence class of codes with the largest possible Hamming distance The largest achievable binary Hamming distance for real self-dual codes with 3.46 W b W 7 = 2 ≤ W d = 11 = 6 2 T-equivalent codes, which are C-equivalent but not T-equivalent with each other. d classes of T-equivalent codes, with the refined enumerator polynomials Lorentzian lattice, whose correspondingSiegel theta-function Narain and CFT hence hasby the ( spectral partition function gap ofthe the term corresponding CFT is given the Construction A lattice ` a twist by ainto half an lattice even vector lattice, when numerical bound on the spectral gapthe is close, Leech but lattice is understood strictly smaller asleaves than a the self-dual possibility Lorentzian for lattice thewith is Leech large odd, lattice spectral see to appendix gap, define a some question special we non-chiral leave fermionic for CFT is the future. The theoretical fit of∆ the numerical bootstrap constraintlattice, for the the unique value self-dual of latticean in the appearance spectral gap when available there we confirm thereof are pairs, more triples, new and examplesn quadruples of of isospectral isospectral codes. There are6.10 instances We have classified all graphs withgenerate up all to correspondingusing refined computer enumerator algebra. polynomials WeT-equivalent leave and codes) the identify with task larger equivalent of ones classifying(classes ELC of classes equivalent of codes) graphs for (classes of 6.9 JHEP03(2021)160 . = I (6.51) (6.46) (6.47) (6.48) (6.49) (6.50) ) with , they T 12 BB 2.15 12 ... y + 12 x 3 ], the permutation q 48 . 12 . ), see figure y + 1296 24 y 10 . If we apply a twist with 6.49 y . Its theta-function, given + + 24 . The Narain CFT defined + 234 14 h τ + 18384512 18 brings it to a form that is self-dual x 10 2 y = 3 − , . } / y 6 , 5 2 2 x 12 q = ` x , 8 0 + 960 , τ 8 . It should also be noted that codes + 64 21 y 2 , / 16 16 22 + 1980 y x , 8 8 23 x y are symmetric but do not satisfy + 1130496 , 4 binary code. It is even but not doubly-even, 8 2 4 = 3 x 17 B q / + 375 , 6] – 63 – b , 6 18 d + 375 y , 12 14 18 , , 19 + 1485 , y x are zero. + 24 code can be brought to the canonical form ( 6 16 10 [24 ∆ = h + 98256 y , ii x + 24 6 = 12020990775258723326 = 8432846454558968306 = 47473099643714589357 . They can be brought to the canonical, or B-form, with 2 B 13 + 64 h / x , 1 2 3 ) along the diagonal 3 k k k q 24 ), 14 = 6 , x +960 b 6.51 15 d + 396 E.1 . Here we give a representative from each class of T-equivalent , B ) = = 24 12 , x d using the generator matrix given in figure 12.1 of [ 20 , xy , = 2 ), is , understood as classical binary code, is the same as the enumerator = 1 + 4096 + 24 11 , one obtains the odd Leech lattice, the unique self-dual odd lattice in ). , h , y W . 2 2 12 10 2.40 x 2 , √ , in which not all ( 3.17 9 / / B , 2 III 7 / , 1 6 Odd Leech W ~ , , we obtain an even self-dual Lorentzian lattice, which, as a Euclidean lattice, ) and reduces to ( 4 = 3 5 Θ ) and ( 2 , = / 1 4 ~ √ b , δ ~ 2 d 3 3.46 Going back to the stabilizer codes from the third class ( 6.50 , If one defines dimensions, with shortest vector of length-squared 2 8 = , 1 from two other classes,∆ = via the same twist procedure, also lead{ to Narainwith theories respect with to ( correspond to an even self-dual Lorentzianis lattice, which, isodual understood and as isospectral a Euclideanδ with lattice, the Construction Ais lattice of isodual and isospectralby ( with the oddwith this Leech lattice lattice. has Its spectral Siegel gap theta-function is given The odd Leech latticemeans that can the be generator of understooda the symmetric as an odd self-dual Lorentzian lattice, which with 24 by ( The odd Golay codewhich is means a corresponding self-dual Construction A lattice is self-dual and odd. Applying twist Understood as binary codes,For via B-form the codes Gray this map,It means these is the codes therefore matrices are remarkable isodualshown that but in enumerator not figure polynomial self-dual. ofpolynomial the of the third odd class, Golay with code the graph All three classes have many possible matrices codes, in the notation of ( Of course all three REPs correspond to the same enumerator polynomial ~ JHEP03(2021)160 ) is .A = 4 b d B.1 = 4 b = d d = ). As a binary d 6.45 , C-equivalence can B in the canonical form ( B ). Qualitatively, classical codes are 4.23 ) with all diagonal elements set to zero B.1 ( (and can be used to construct a Narian CFT B . Self-dual quantum stabilizer codes correspond – 64 – = 6 3.2 b d . Via the twist construction it gives rise to a non-integer + 24 h and = 4 d ), a representative from one of the T-equivalence classes associated with . This means that the Golay code can be interpreted as a self-dual 6.49 6= 0 ii B via twist construction), and other two have a more modest ): one has 2 7 / . Graph of ( ∆ = 3 Our main focus has been on self-dual codes and lattices. The space of stabilizer codes is Finally, returning to the Golay code, its matrix for which this is possible. codes correspond to Lorentzian lattices and non-chiral CFTs. discrete, but in our constructiondual it lattices. is embedded This in isspace the an of continuous essential space Euclidean difference self-dual of from lattices Lorentzian the self- is case discrete. of Within classical the codes, Narain moduli for space which of the all self- lattices, which is the subjectto of self-dual section lattices, and realCFTs based codes on to even even self-dual lattices. lattices,CFTs are which In captured we this by call way, the code real corresponding theories.given self-dual codes; by codes Basic in the properties define particular, code’s of the refined code CFT enumeratorrelated partition polynomial to function ( Euclidean is lattices and chiral CFTs. In this paper we have shown that quantum 7 Conclusions In this paper weclass have discussed of a quantum relation error-correcting betweenkey codes, quantum ingredient stabilizer in and codes, our a a construction class particular is of the 2d relation conformal between stabilizer field codes theories. and Lorentzian The classes mentioned above. This seemssimply to remove indicate all that non-zero forThis diagonal this is matrix matrix an elements while unusualB situation, leaving everything and else it intact. would be interesting to describe the class of matrices self-dual codes. There aresee three footnote classes ofwith T-dual codes (strictly speaking,curious at observation least here three, is thatgives the rise matrix to the stabilizer code, which shares the same REP with one of the isodual code itisodual is lattice isospectral isospectral with with the odd Leech lattice. symmetric but stabilizer code, which is not real. We can use code equivalence to find C-equivalent real Figure 12 the dodecacode. The corresponding refined enumerator polynomial is given by ( JHEP03(2021)160 8 ’s ) ≤ ¯ τ . n τ, 8 ( , which ) Z ≤ ¯ τ n τ, .) ( -dimensional = n Z c 2.1 WZW theory in , understood as a 1 8 E \ . SO(8) reveals a growing number , see section . A drastic reduction of the 6.10 8 12 ≤ 6.10 n . Finally, we constructed an isodual . Using T-duality transformations one , and which is E 8 D divisible by E . Other special lattices, in particular the 24 6.4 ≥ characters with positive integer coefficients (the – 65 – c n can be interpreted as the graph adjacency matrix. U(1) B × -flux, such that it is fully specified by a binary symmetric ; it corresponds to a pair of isospectral even self-dual n B nodes, see appendix = 7 U(1) 8 n ≤ , which suggests one of two possibilities. It could be that these . This theory is based on the root lattice of n n . At the level of graphs, informed by the CFT interpretation, we . The matrix = = 4 5 c n = mod 2 c B ), yet do not correspond to any known theory. The number of fake 1 ). There are other spin-off results which can be formulated without references B = ’s might correspond to actual CFTs from some new sector, most likely related to 4.36 ) ¯ τ Finally, our analysis of stabilizer codes with small Schematically, one can think of code theories as a particular “ansatz” which reduces Code theories form a subsector of Narain CFTs. T-duality transformations can map a τ, ( “fake” chiral CFT partition functions for of isospectral butof physically view inequivalent these Narain aresuch CFTs. examples example of appears From isospectral the for but mathematical non-isomorphic point Narain lattices. The first are not partition functionsmodular of bootstrap any exclusion actual plots CFT, inZ fact which might means be persistent empty. Another alloweda possibility regions is family in that of these (non-additive)be codes. extended This to include wouldcompletely mean these analogous and that construction perhaps the exists other notion for sets of classical of a binary theories. code codes (We CFT leading also could to mention examples that of a modular invariance constraints at the levelof of code “fake” theories CFT gives us partition aadmit multitude functions, expansions of in modular examples terms invariant of non-chiralfirst functions being quickly grows with particular code theory whichdisguise, we dub attains non-chiral the maximalcentral value charge of theLorentzian spectral even gap self-dual among lattice,odd the see Leech Narain section lattice, theories also with make an appearance, see section modular invariance of the CFTby partition a function multivariate to polynomial. aseveral simple questions algebraic In central condition this to satisfied wayclassical the binary code conformal codes modular theories which bootstrap provide give program. a rise playground to As to optimal in sphere probe the packings case in of certain dimensions, a cube of “unit” size and quantized matrix Thus code theories can beby edge labeled local by complementation. graphs, By with classifyingnodes graphs all we classes of have of T-dual ELC found equivalent theories all graphs being on physically related distinct code CFTs with central charge classified all graphs on non-integral lattice isospectral to odd Leech lattice in section code theory into another codeequivalent theory, in in the which case code the sense, corresponding ascan codes proved always in are appendix bring necessarily any code CFT into the form of a compactification on an coset ( to CFTs. We havecanonical derived form, and the calculated Gilbert-Varshamov linear bounddistance, programming bounds by see on averaging section the over largestoutlined binary all Hamming the codes importance in of edge local complementation (ELC) equivalence classes, and dual even Lorentzian lattices, we describe the set of real self-dual stabilizer codes as a group JHEP03(2021)160 ], 100 ] or re- 15 lattice CFTs 2 Z / dimensions, there are 24 Spin(32) and 8 E that the ensemble average over all ]. Another question is the recently ]. In light of our work, one of the × 5 8 20 98 , E theories, with the number presumably 2 , which can be effectively studied using b ]. We expect the construction outlined in d = 8 5 , n 3 = – 66 – c . For chiral CFTs based on Euclidean lattices, the ]. At the level of code theories, the analog of the 10 10 , 9 [ . 1 n  ], quantum information properties of CFT ground states [ = c n ]. We have argued in section , see figure 99 7 , 7 = 17 , R c 10 to Lorentzian lattices with large shortest vector. To that end one needs to go b d Let us conclude with one more fundamental question: to what extent does the physi- There are several different ways in which classical codes may be associated with various Code CFTs may provide a useful framework for addressing the following two ques- Acknowledgments We would like to thankand Noam Eric Elkies, Rains Antonunder Gerasimov, for Nikita Grant discussions. Nekrasov, No. Vasily PHY-1720374. Pestun, AD He is would supported like to by thank the IHES National for hospitality Science during Foundation the cal Hilbert space ofinherited, a from code the theory associated exhibit“quantum codes? quantum error error-correcting correction” Here necessitated properties we by related, havelated the or to in emergence the of mind large locality various N in properties,and limit the [ probably including bulk many [ others. associated with the Leechbeyond lattice, pure and geometric its Monster symmetriesimmediate orbifold, of questions exhibits the would be symmetries lattice to which [ code study go symmetries CFTs, of possibly the leading non-chiral to VOAs associated a with non-chiral moonshine theory. to non-chiral CFTs. We alreadynon-real mentioned stabilizer a codes, possible associatedCFTs. connection with between self-dual self-dual But odd albeit we Lorentzianimportant lattices, expect aspect and of that fermionic theformer many relation can other be between constructions used Euclidean to are lattices define and possible. consistent Vertex chiral Operator CFTs Algebras Perhaps is (VOA). the Thus, that the most the VOA suggesting a holographic interpretation.testbed to Thus, verify and code study theories this may duality. provide anchiral CFTs, additional both supersymmetric andthis not paper [ to be perhaps the simplest but not the only scheme relating quantum codes beyond Construction A lattices,constructions B, discussed C in etc. this developedproposed for paper, holographic classical duality and codes between introduce averaged [ theories Narain some in theories the analogs and bulk certain [ of Chern-Simons code theories exhibits the same basic features as the average over full Narain moduli space, tions. The first isNarain to theories understand with the asymptoticspectral behavior gap of the is maximal thelinear spectral programming binary gap methods. Hamming for It distance large is an open question though to relate quantum codes with lowest-dimensional pair of isospectral CFTscorresponding are the to Milnor’s exampleIn of contrast isospectral to even the self-dualmany Euclidean lattices dozens case, in of where 16 next examplesgrowing dimensions. example of rapidly occurs isospectral for in larger Lorentzian lattices in JHEP03(2021)160 8 n E code (A.1) (A.2) (A.3) . We 4] , , we can 4 , [8 2.1 , 8 we also discuss are isomorphic E 8 8 E E = 0 (mod 2) are simultaneously i i x x and i 7 . P E                                 1 1 0 0 0 0 0 0 1 0 − − where all 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 ) n code and the extended − − − − − − − − − , ) 1 0 0 0 3] δ , − 1 0 4 + this lattice is the root lattice of the Lie , − 1 0 01 0 2 0 1 0 1 , . . . , x n 1 [7 does not belong to the lattice, but when − − 1 0 0 D x ( ( = 8 − 2 1 2 / ∪ − 1 0 0 0 n 1 ~ – 67 – n − 1 0 0 0 0 0 0 = D − 1 δ ~ − = 1 2 1 0 0 0 0 0 0 00 0 00 0 0 0 1 0 0 2 0 0 1 0 0 2 2 0 10 2 0 0 0 0 ). For − 1 1 0 0 0 0 + n 1 0 00 00 0 0 0 0 1 00 0 00 1 0 00 0 0 0 1 0 0 1 0 0 1 0 0 −                 D as it is more symmetric and simpler. For series is the “checkerboard” lattice of integer vectors 2.38                 = 8 8 = n E E 8 D Λ E 8 . The vector Λ n T E Λ D with the sum of all coordinates being even, lattices and codes n 8 Z is defined as in ( E δ ∈ does. Using a procedure similar to the twist described in section . It includes vectors of the form ) + 8 n δ and n 8 2 E 7 D E E , . . . , x 1 . We start with the case of 8 x as a generator matrix, in which case gram matrix is the Cartan matrix of algebra either integer or half integer, and their sum is integer and even. One can choose define a new lattice where A.1 The root( lattice ofdenote the this lattice as is even, In this section we showto that the the Construction root A lattices lattices ofe of the the Lie Hamming algebras the equivalence of different Lorentz-signaturetheory metrics to and the the relation theory of of the eight non-chiral free Majorana fermions. the European Union’s Horizonagreement 677368). 2020 research and innovation program (QUASIFT grant A visit, which was supported by funds from the European Research Council (ERC) under JHEP03(2021)160 , ) is Z 8 , = 2 ), E (A.6) (A.4) (A.5) 2 ` A.4 GL(8 . Curiously 1 such that their scalar follows from ( 8 calls the task of finding . . The lattices generated O E 8 2 code. The latter is given e Λ √ Λ 4] / by an appropriate ) are found, one can choose , 4 8                 . , E A.5 1 0 0 0 0 0 0 1 [8                 Λ − is known, Wikipedia roots of roots, i.e., vectors of length 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 Z − − − − − − . 240 1 2 1 2 8 240 2 1 2 1 2 1 2 1 2 1 2 1 e − − Λ 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 . Once O − − Z = 1 2 1 2 – 68 – 2 1 2 1 2 1 2 1 2 1 2 1 0 1 1 1 1 1 1 2 2 0 0 00 1 2 1 0 1 0 0 0 1 0 1 2 0 0 1 0 1 0 0 0 1 2 1 1 0 1 1 1 1 0 1 1 2 1 1 0 1 1 1 1 1 0 2 1 1 1 1 1 1 2 1 Z − − 8 ); we will denote it by                 E 1 such that = Λ − ) ) is difficult; indeed, 2.18 8 Z 1 0 e , − Λ A.4 is of course very different from the generator of the Con- 8 1 0 0 T e 8 0 10 0 1 0 0 0 00 0 0 0 0 00 1 0 roots from the list of as in ( − E Λ GL(8 8 Λ                 B ∈ associated with the Hamming = ) ). The procedure is iterative. Using computer algebra we calculate Z 8 e O A.5 is even and self-dual, which follows from all diagonal matrix elements Λ( are not identical but are isomorphic, which means there is a rotation 8 directly from ( E 8 Z e matrix of scalar products. The first root is chosen at will. We choose the Λ and 240 and a matrix and O ) with the matrix × ) being even, while the matrix is integer and has determinant 8 E O(8) are roots. We consider the Gram matrix The lattice Once a set of roots with the scalar product matrix ( The following trick saves the day. There are The generator matrix 2.17 240 A.5 Λ 8 e of ( not be able tolattice find has a many symmetries vector it with works the well desired in properties practice. them at to a generate certain thetransformation. step), That lattice, but is since which the the will desired matrix be related to product is given by ( the second root from the setThe of third those is which then haveproduct chosen the with desired from the scalar the first product list two, with of and the those so first remaining on. one. roots The which procedure have does desired not scalar guarantee success (we may Our goal now is to choose Finding the explicit isomorphism “not entirely trivial.” which can be written explicitlyΛ in both representations. In particular all of the columns of struction A lattice by ( by O ∈ JHEP03(2021)160 L p (A.8) (A.9) (A.7) to the (A.12) (A.10) (A.11) 8 E ) we used ? It turns Λ η 6.4 g . (In physics 8 g T E is even self-dual = Λ 8 e , Λ ). Indeed, g T is also even self-dual       T 8 , T ) E 3.17 satisfies 1 1 , ” Narain CFT. Z Λ g to make it antisymmetric. − 8 , O E 1 0 0 B T − = GL(8 1 1 1 0 0 0 0 0 0 1 1 S − − ∈       ,                 , would be an orthogonal matrix from 2 H = ) 8 √ R . e R / Λ O O ! g S. g is a symmetry of I × O T T − L T S O – 69 – , II I O ( = =

      η H η 1 = is in the part of the T-duality group that rotates block matrix = − 0 0 00 0 0 0 0 10 0 0 0 1 0 0 00 1 0 0 0 0 1 10 0 1 1 1 1 0 01 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 0 1 0 1 0 8 R H . We leave the exercise of calculating ). Then 1 1 1 T g                 × − − and 8 = 1 0 0 1 0 0 × O 4.10 g 8 − − L e O 1 Λ 0 0 0 0 1 − g 8       T e and the Λ ) = R L , O O(4 ∈ , which guarantees that the lattice is even and self-dual.) In section ( ) L,R separately.) Accordingly, the orthogonal matrix R O , R 4 p An immediate check reveals that the lattice generated by , understood as a Lorentzian lattice to define a “non-chiral 8 It turns out that for the particular choice of terms, the transformation and where performs the transformation ( One can immediately ask,out what that is this is theorthogonal exactly Narain transformation the CFT of same the defined theory form with thanks respect to to the lattice symmetry. We consider an reader. This is curious, becausewith it respect means to that both the metrics, lattice generated by Gram matrix are even.The (Alternatively lattice one would can remainO(4 flip the signs same, in butE now with respect to the same metric from which follows that it is self-dual. It is also even because all diagonal elements of the also even and self-dual with respect to Lorentz signature metric ( JHEP03(2021)160 ] with (A.15) (A.16) (A.17) (A.13) (A.14) 101 coincide, leading to 4 theory and the theory flipping the sign of one D 8 R E , are T-dual to each other. Bγ O 4 η , , understood as a Lorentzian T . D ) ) 2 or γ 8              Z e √ and L 1 , / 1 0 0 0 0 0 g − O Λ( . and ! 2 1 2 1 2 1 2 1 2 1 2 1 2 4       4 1 4 √ GL(8       D − − − − − − D D — the root lattice of SO(8) [ γ ∈ γ 1 1 4 4 Bγ 1 0 1 0 T S D − 1 1 D 1 γ − − − 1 1 1 − ) 1 − 1 1 0 4 0 − ) with either T − D 1 1 0 0 – 70 – ]. The resulting Lorentzian lattice with the gen- 1 0 00 0 0 0 γ ,Z 1 0 1 10 0 0 0 0 0 0 − 1 1 0 0 S 2( A.9       −       Z 104 8

– 1 1 0 0 0 e = 1 0 00 00 0 0 0 00 0 00 0 0 0 0 1 0 0 = − = Λ 4 B 102 1 D              = γ 8 = e lattice has a rich group of symmetries, most of which act \ 7 SO(8) Λ E 8 Λ S Λ E WZW theory as a Narain CFT. It is related to the lattice generated can be defined via the generator matrix 1 7 E global symmetry [ \ SO(8) SO(8) 7 by a T-duality transformation ( E × 8 e Finally we discuss the equivalence between the non-chiral More generally, the Λ is a symmetry of the lattice, A.2 The root lattice describes the by arbitrary coordinate. is chosen such thatSO(8) upper triangular partserator of The B-field generator matrix nontrivially on the Lorentzianwith metric, respect “rotating” to it any into choice a of the new Lorentzian one. metric are Narainof physically CFTs eight equivalent defined to free each fermions other. leading with to the a diagonal toroidal GSO compactification projection. CFT The on fermions can be bosonised, Therefore the Narain CFTs correspondinglattice to in the two lattice different ways, with respect to the metrics S JHEP03(2021)160 (A.20) (A.21) (A.18) (A.19) ). The roots of 2.7 126 . After that an 7 e , we will employ a Λ 7 E Λ . Then we calculate the (and permutations), and 4 . 2              √ 1 / ) Lie algebra, − 6 . . 0 7 1 0 2 , E              2 − √ / ± 1 0 1 2 0 code is the transpose of ( ( ,              − − 1 1 0 0 1 0 0 4] − 1 0 01 0 2 1 0 0 2 , − − Z 3 − − − 8 , 2 0 0 0 e 1 0 01 0 2 0 [7 √ Λ − − − O 1 0 01 0 2 0 0 – 71 – 1 0 0 0 1 − − = with the one generated by − 8 0 0 0 0 1 1 1 2 0 0 00 1 2 0 0 0 00 0 0 1 2 0 00 1 0 1 0 0 20 0 0 0 0 1 00 1 0 0 0 1 0 0 1 1 1 2 7 2 0 0 0 0 00 0 00 0 0 0 0 E 1 0 0 0 1 0 e −              Λ − Λ codewords of Hamming weight              = vectors of the form 1 0 0 0 1 0 00 0 00 1 0 0 0 10 0 00 0 1 0 0 0 0 1 0 7 7 = − e which expresses those roots in terms of 14 7              Λ ) E Z Λ , 7 T E ). The process does not need to succeed and in practice we needed Λ satisfying GL(7 A.18 ∈ O 1 − Z scalar product matrix, and start choosing roots one by one such that their scalar vectors obtained from the 126 7 The generator matrix of the Hamming × × 4 can be found easily to experiment with a fewbe different completed. candidates Once for those the rootsfind fifth are a vector, identified, matrix before we the canorthogonal process solve matrix a could system of linear equations to procedure analogous to thethe one code lattice, used which in include 2 the previous section.126 We construct product is equal to ( To match the lattice generated by generator matrix of the Construction A lattice of this code can be chosen as such that the Gram matrix is the Cartan matrix of the JHEP03(2021)160 (B.2) (B.1) ) with the symmetry). 4 2.39 , it would lead to U(1) would immediately is an even self-dual . Alternatively, one × = 4 ) 2 4 2 ) 24 √ ` g / 24 2 g GF(2) Λ( / U(1) 1 . ~ Λ( symmetry) is strictly smaller                           = 12 δ ~ being the optimal sphere packing . Here we provide an independent , 8 2 U(1) by applying a twist ( E 12) √ × ) , / , provided it could be reinterpreted as a 12 24 ! g (12 T can be defined using a generator matrix of lattice can be understood as a Lorentzian I Λ( B U(1) 8 . If we could interpret it, via the Gray map, , understood over 24 ∆ = 2 E g 24 0 g – 72 – 2 I = I . Given that the Leech lattice yields the optimal (and

T = = 2 BB 2 = 12 24 ` g that the n 010101010111101010101011 010011111010001001111101100100111110110010011101111001001110111100100101111110010010011111001001001111100110 100111110001 Λ Golay code                           8] A.1 , 2∆ = 12 B = , ) with [24 is integer, unimodular, and with even diagonal entries. dimensions with the shortest vector of length . for the given value of central charge (and 2.15 8 theory with spectral gap Λ . 24 E 2 T √ Λ / = 12 2 ∆ = 1 c / 1 ~ ]. This indirectly proves that the Leech lattice is not an even self-dual lattice in = 10 [ δ ~ for any Lorentzian metric with signature 2 Euclidean dimensions, with a spectral gap specified by the maximal possible length of Our starting point is the Golay code The Leech lattice can be obtained from We have seen in section 12 , 8 12 and more explicit demonstration ofLeech this fact, lattice underscoring and a notable difference between the as a self-dual stabilizerLorentzian code, lattice. it would Then immediately applying follow a that twist with the same sphere packing in a non-chiral Lorentzian lattice. It has beenspectral recently gap shown for using all numerical theories modularthan with bootstrap that the R even self-dual lattice. Itspectral can gap be used toThis define extremal a property non-chiral can CFTin be with traced the to largest the possible the lattice shortest lattice vector, and check that vector This defines a self-dualcan code define since the generator matrix of the Construction A lattice The binary extended the canonical form ( B Golay code and Leech lattice JHEP03(2021)160 ) ) d B.1 O( (B.3) (C.2) (C.1) trans- × ). Next ) . It can and last 1 p ) is nonde- i k ~ d symmetric u O d, d 11 . It is a null B B O( b 1 is an integral O( by u I , understood as → 24 ) , g B 24 Λ ! , which is satisfied, and moreover g , as well as (compare g T | p -dimensional vectors, 12 Λ( 2 R d b 24 O k T g ~ 1 | B Λ − 11 b 12 = . b B = B | → 1 ij 21 2 L δ b − 11 B k ~ b | + of the binary Golay code as the = by an appropriate ). Let’s start with j 22 I ˜ u B) b ) that the sub-matrix · | C.1 i 1 . B.3 − 11 1 2 R b − 11 k ~ = ( I b , u · 21 b T (which will be denoted simply as 1 k ~ G

1 L = 0 k ~ j = = satisfying ( – 73 – ˜ u ), which states that any Narain lattice can be 0 2 L · j k ~ B i ˜ u will have the same length. Using a transformation · include permutations ˜ , u 4.21 i 1 . For the same reason 1 R B u k ~ ) k ~ , → 2 R k ~ vectors (columns) of the generator matrix , . It is assumed in ( ) respects this property. Therefore if we hope to bring ( = 0 2 L and d ! j k ~ , we must do it solely using permutations to be equal to 1 vectors L u B.3 22 12 · k ~ 1 b R d b = ( i ,( ). . They satisfy GF(2) k 2 ~ = 0 i u 2 ˜ u ~u ii 11 21 = 0 b b 4.11 B ii ,

T B is not necessarily symmetric and may have non-zero diagonal elements. transformation leaves the metric invariant, we can say that an arbitrary , and therefore if we represent it in terms of two B , which is not. In other words, understood as a stabilizer code, the Golay ) B = and , the vectors = 0 ) T = 0 B = B d, d . The transformations of 1 R 2 we can rotate | ii )) k ~ 1 is generated by O( ) , B u d = 0 1 L | Λ k ~ B = B 4.39 ii -twist would yield an odd self-dual lattice. O( δ Our starting point is equation ( The matrix To summarize, the Leech lattice, considered as a Lorentzian lattice, is self-dual and odd. One may wonder if one can use code equivalences to define a new code with B = ( vectors (columns) by 1 from we consider the vector Since an lattice vector, brought to the form ( obtained from the cubicformation. lattice Let with us generator denote matrix d first C Any Narain CFT isWe a want toroidal to compactification showtransformations, any that, even using self-dual Lorentzian symmetries lattice of (a the so-called Narain physical lattice), theory, can namely be But if to the form be easily seen that this is impossible. where all algebra isgenerate. over unimodular matrix with oddthis diagonal entries). Proceeding to define the Leech latticeand via with ( would like to interpret thegenerator generator matrix matrix of abut real also stabilizer code.code For is that self-dual we buta need not Lorentzian real. lattice, Therefore is the self-dual corresponding but lattice odd (one can check that yield the Leech lattice, now viewed as Lorentzian even and self-dual. In other words we ~u JHEP03(2021)160 ∈ = R 2 L (C.7) (C.3) (C.4) (C.5) (C.6) k (D.4) (D.5) (D.1) (D.2) (D.3) ~ × O L we obtain i O ˜ k ~ , and after a , I . i , and the anti- ) k − ~ ∗ R Γ = ), we can make O which would map the n, n, 2 1 k ) ~ 3.19 Q O( n . An element 0 ∈ C O( . . ! ) × ) i ) R R ˜ k Z ~ . , ) n O i i . ˜ k ˜ k ~ ~ satisfying . 1 , but in this case an orthogonal , which is dual to O( − O + i , define a lattice, which we can take − i β n, n, . ˜ β u ) i L L Q ˜ k ~ Γ = ( ij , ˜ k 2 ~ T − 2 + O O i δ ˜ Q u    ~ α = ( α . = 1 n i R R ) for some code R Mat(2 L + i j β ˜ β u ~ ... ) O and a trivial redefinition of k ~ k 0 ~ ∈ Q , ) ) C i , 1 n + − O ˜ T )( d ) k → O G ~ → O α α ... i L L T Λ( Q k ~ , – 74 – β O( β 1    O O ) form lattice = ( i + − 2 2 − i i × k + ~

are equivalent (they define the same code and the = . The vectors k ~u ~ ) Q i α O 1 2 ) α ˜ ij ( T d G, Q − Q , −Q δ · T i , , β G = = i = i i k ~ O( Q = ˜ k k ~ ~ + 2 = ( α R L will bring them to the form Q j + ~p Q ~p = ( i ) is G ˜ u B ) is given by i as follows ~u · Q ~u ∼ i = ( 4.8 D.2 G u i G = ( ~u satisfy Q × Q ∈ G, i i ~e k Q ~ from ( = 0 B into another code lattice G . We can find an orthogonal matrix ) ) C d → would act on Λ( ) O( G n . The vectors . Continuing this logic, we find O( ∗ 2 O ∈ Γ k ~ × ) = n 2 R Another way to represent ( We also remind thesame reader lattice) that upon shifting components by even number, we would like identify allcode possible lattice transformations from O( Starting from a particular code CFT with code generator matrix ( Therefore, the vectors symmetric matrix D T-duality as code equivalence The last step isto to be impose where diagonal transformation k We can repeattransformation the acting same on the procedure for the vectors By an orthogonal transformation in the directions orthogonal to ~ JHEP03(2021)160 , , ) ) w 2 p n are O ∈ (D.7) (D.8) (D.6) ) is a O( w × O n × . Going for ∈ 2 p p ) are even, ≤ i n O k l O i ν , and reshuffle I)( O( . ≤ k L,i y p ∈ ), which is too × p n, σ O × O 1 O is equivalent. p p can be shifted by are odd, then the does. In this case ≤ or R O D.8 i i l i ( I g ν O ×O α, β p ≤ O 1 , are code equivalences, , all il i must always be integer or must only reshuffle integer δ , p R ik k or with i δ O β 2 , p L , provided (applied to all L 2 j ) + are of the form O − p i g Z . For simplicity we can assume k i g β kl ... α w δ n, , all components of to correspond to a code lattice, the ) ⊗ which transforms a code (lattice) into i = = O( R i and k I matrices are either ν , while , β . It is easy to see that the simultaneous which are reshuffled by i i × k L,i , p I kl n σ × k ) α y α L ) can be represented as p i Z ( × σ p Z 2 p O O i ∈ ( or n, O a , w, p I ) , – 75 – × O O( ⊗ · · · ⊗ n ∈ a 1 p i 1 ∈ acts on it trivially. Otherwise, when all ν O O( σ + 2 ) and taking into account that p k, l R i × codewords α O  ) such that first D.5 n invariant, the permutation that simultaneous permutations = n × O i → with each other. In other words, provided there is a subset i must be simultaneously integer or half-integer. Since the g R and L ), ( g L are either zeros or ones, which are reshuffled by α O( p 4.2 ) p R O p ∈ L,i D.4 ), in the first case the generator remains invariant, while in the i are either even or odd. If, for a given ~p ,... mod 2 for } 2 k leaves and / D.7 matrices are either . The diagonal part has been already discussed, and we only need 1 × O l L,i would belong to the code, since the code is self-dual. We therefore 1 ≤ I) L k p I − i l p . In order for the vectors ) indices, in which case all generators × g k {z 2 p p I) of a particular component of ≤ k = 2 ,..., O O i 1 such that for all 2 ( × ( / 1 p k L,i } p | p O for O × O 2 ( i l components of i = (1 = ν O is an arbitrary permutation of indexes within k p acts on all codes by transforming them into equivalent codes. Furthermore, if a l,i , . . . , n p p O ) 1 O Z So far we have only considered the transformations of the form ( To summarize, we have shown that any transformation of the form If we now take a particular A pair of arbitrary permutations Next we consider sign flips of the form We already saw in section n, ⊆ { -th components of transforms a given codelence into sense. another code, the codes arerestrictive. equivalent in Going back the to code ( arbitrary equiva- even-valued vectors, O( transformation them, the new vector willthe trivially new commute reshuffled with allconclude the that any transformation ofanother the code form (lattice) is in fact a symmetry of that code (lattice). where all vector the first back to the generator ( second case the first then includes the first w simultaneously integer or half-integer, where to analyze i transformation or half-integer components of as well as signmapping flips codes to equivalent codes. sign flip symmetry of the lattice. Therefore flipping the sign with JHEP03(2021)160 . , ) ) 2 / Z is a 768 D.9 is an (E.1) (D.9) k, kl ) = 1 L -th row ) we find possible O( | k and that O 6 L,R kl , from ( D.9 are integer 2 ) O 0) ( , R 0 L,R , O 0 ) = I H , i , ( . ) belong to the code | α 2 2 ( / and ) can be parametrized √ = (1 i is degenerate. This would be one of b L − would not change the 0 i B block, which, without n mod 2 R O symmetric matrix with α = ) considered above. In . Finally, if 4 G H a  n R 1)(2 × B − O D.8 i × ( 4 R − n H matrices and is an integer it must be equal to +1 i 2 . Combining all this together and − 4 upper-left block, kl + -th column must be zero. Because 2 , an j l ) L we can scan through all / , which maps codes into equivalent 4 × − B ) (upon rescaling by 1 H 2 R 4 Z / L,R × 4 ± happen to be integer-valued (which is Z H 2) 4 O n, ( I + − 0 i ∈ must be integers. Therefore, if or (ii) an orthogonal matrix from n α such codes. R O( nodes 2 8 Q a, b H / +1)( 2 1 8 2 n , sign-flips and row operations. Sign flips can be + ( using the following non-degenerate map for ± – 76 – p O ∈ 2 L ≤ ) are connected. All matrices 1 , ij H j -th row and the n ×O − a, b B must be even. Indeed, by taking  k p are orthogonal 2 is even. Applying the same logic to the inverse of ( / +1 0 O B) and i is a half-integer, so must be n R 1) | i O × O X ) = = to be block-diagonal matrices where each block is either , β 0 i − ) = 2( L 0 j and there must be three other components in the (I has at least one half-integer n are orthogonal, if α ,H α ( into the Hadamard matrix, while O ( 2 n L n =1 define a new code, the generator matrix can be brought into α, β L,R i X / 2 ( L which are also equal to 0 i H is nondegenerate. Similar logic with sign flips and permutations 1 must belong to a code. Any real self-dual code which includes the codeword L,R L,R O to define a code lattice), as a matrix H = ) 1 ≤ ~ ± kl 0 i , β O O must have the same block structure. is even. There are only 0 ) → 0 i k k α combined with reshufflings of columns of G α ( R , and if ), all components of and permutations won’t change the canonical form; we therefore can L,R ≤ p O R α, β , without loss of generality we can assume that R O 0 . Here if the vertices O ↔ i ( G D.3 β × O ) mod 2 = ,H ) = I and i a L p = 1 to conclude that whenever all L α L B ( H O H ij 9 O has no half-integer blocks, this is the case of ( 2 , B B a, matrix with all elements being L is a symmetric matrix, and focus on the 4 L,R H and O × (2 . Both -th column of 4 l n If = 0 Since all vectors of the form , and all other components of the ) = B 9 i ≤ 1 ii is even, hence the code defined by β lattice, it can be shownwe that find the code 1 that the code defined by We can parametrize graphsB by an adjacency matrix by integer numbers choices of necessary for theconcludes new our proof. E Classification of graphs on canonical form using permutations absorbed into assume that the matrix can be used topossible transform combinations of signs. For each choice of and similarly for Provided that the new what follows we assume loss of generality wepermutations can assume tocanonical be form located of in( the upper-left corner. Since diagonal and using a diagonal transformation codes, we can always(i) take a k or half-integer. Since the ± of the symmetry ( integer, so must be half-integer, it is equal to half-integer, we immediately conclude that all matrix elements of ~ JHEP03(2021)160 nodes. 12 , A000088 8 available . For each ≤ ) among all components, 2 JHEP (1984) 3256. n 8 , E.1 (2000) 1382. ( ≤ 81 (2000) 1238. graphs8 k , 47 1 47 is a list with each entry . Integer sequences ]] number I n j i , Not. AMS i . , , Not. AMS . n , [[ , and the correct number of graph 1 maximal Proc. Natl. Acad. Sci. of which correspond to decomposable (1996) 61 , (number of graph isomorphism equivalence (1998) 407 I n LC n t i ELiELCiI A natural representation of the fischer-griess 179 193 Vertex operator algebras and the Monster , see table ]. Conformal field theories, representations and nodes Sphere packing and quantum gravity Framed vertex operator algebras, codes and the – 77 – ELC n n which is a nested list of lists. It has i as character j Holomorphic SCFTs with small index SPIRE IN and ][ ]. ), which permits any use, distribution and reproduction in . ELC n corresponds to a particular ELC equivalence class within t ELiELCiI 3 elements, the first ]] correspond to indecomposable graphs, see table i , SPIRE LC n n Commun. Math. Phys. t IN LC n [[ [ , Commun. Math. Phys. i , − CC-BY 4.0 Lattices, linear codes, and invariants. Part II This article is distributed under the terms of the Creative Commons LC n Lattices, linear codes, and invariants. Part I . Number of inequivalent indecomposable graphs t arXiv:1905.01319 [ 12 ELiELCiI is a list of ≤ , correspondingly. ]] n n LC n [[ t . Number of inequivalent graphs on 1 2 3 4 5 6 7 8 9 10 11 12 1 2 4 11 34 156 1044 12346 274668 12005168 1018997864 165091172592 1 1 2 6 21 112 853 11117 261080 11716571 1006700565 164059830476 ≤ arXiv:1811.00589 (2019) 048 lattice constructions moonshine module monster with the modular function Academic Press, New York U.S.A. (1989). th LC equivalence class. Each element of i . It contains one variable N.D. Elkies, N. D. Elkies, T. Hartman, D. Mazáč and L. 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