Conference Matrices and Unimodular Lattices

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Europ. J. Combinatorics (2001) 22, 1033–1045 doi:10.1006/eujc.2001.0539 Available online at http://www.idealibrary.com on

ROBIN CHAPMAN

Conference matrices are used to deﬁne complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice. c 2001 Academic Press

1. INTRODUCTION

We use conference matrices to deﬁne an action of the complex numbers on the real Euclidean n vector space R . In certain cases, the lattice Dn+ becomes a module over a ring of quadratic integers. We can then obtain new unimodular lattices, essentially by multiplying the lattice Dn+ by a nonprincipal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the Leech lattice, can be constructed in this way. Recall that a lattice 3 is a discrete subgroup of a ﬁnite dimensional real vector space V . We suppose that V has a given Euclidean inner product (u, v) u v and the rank of 3 equals the dimension of V . In this case 3 has a bounded fundamental7→ · region in V . We call the volume of such a fundamental region (measured with respect to the Euclidean structure on V ) the volume of the lattice 3. The lattice 3 is integral if u v Z for all u, v 3. It is even if u 2 u u 2Z for all u 3. Even lattices are necessarily· ∈ integral. The lattice∈ 3 is unimodular| | =if 3· is∈ integral and has∈ volume 1. It is well known [9, Chapter VIII, Theorem 8] that if 3 is an even unimodular then the rank of 3 is divisible by 8. For convenience we call the square of the length of a vector its norm. The minimum norm of a lattice is the smallest nonzero norm of its vectors.

2. CONFERENCE MATRICES

Let l be a positive integer. A conference matrix of order n [7, Chapter 18] is an n-by-n matrix W satisfying (a) the diagonal entries of W vanish, while its off-diagonal entries lie in 1, 1 , (b) WW (n 1)I , where I denotes the n-by-n identity matrix. {− } > = − Let n denote the set of skew-symmetric conference matrices of order n. W t 2 Let W n. Then H I W satisﬁes HH (I W)(I W) I W ∈ W = + = + − = − = I WW > nI . As all the entries of H lie in 1, 1 then H is a Hadamard matrix. Consequently+ = [7, Theorem 18.1] n 1, 2 or is a multiple{− of} 4. = n Suppose that n is a multiple of 4 and let l n 1. Fix W n and let V R denote the n-dimensional real vector space under the standard= − Euclidean∈ W dot product. We= shall make V into a vector space over the complex numbers. Each complex number z has the form r s√ l where r and s are real numbers and √ l i√l. We then deﬁne + − − = r s√ l v v(r I sW) rv svW + − = + = + for each v V . Then, since W 2 l I , it is straightforward to show that V is now a complex vector space,∈ that is = −

0195–6698/01/081033 + 13 $35.00/0 c 2001 Academic Press

1034 R. Chapman

(a) z(v w) zv zw for z C, v, w V ; + = + ∈ ∈ (b) (z1 z2)v z1v z2v and (z1z2)v z1(z2v) for z1, z2 C, v V . + = + = ∈ ∈ As a complex vector space, V has dimension n/2. Let v √v v denote the Euclidean length of a vector v Rn. This action of C on n | | = · ∈ R transforms lengths in a simple way. Let z∗ denote the complex conjugate of the complex number z. n LEMMA 2.1. (a) If z1, z2 C and v1, v2 R then (z1v1) (z2v2) (z1z2∗v1) v2 (b) If z C and v Rn then∈zv z v . ∈ · = · ∈ ∈ | | = | || |

PROOF. Let z j r j s j √ l with r j , s j R. Then = + − ∈ (z1v1) (z2v2) (z1v1)(z2v2)> · = v1(r1 I s1W)(r2 I s2W)>v> = + + 2 v(r1 I s1W)(r2 I s2W)v> = + − v((r1r2 ls1s2)I (s1r2 r1s2)W)v> = + + − (z1z∗v1) v2 = 2 · as claimed. Consequently

2 2 2 2 zv (zv) (zv) (zz∗v) v z v v z v . 2 | | = · = · = | | · = | | | | Thus for ﬁxed nonzero z, the map v zv is a similarity of Rn with scale factor z . 7→ | |

3. QUADRATIC FIELDS

We retain the previous notation. Suppose in addition that l n 1 is square free. Let K denote the quadratic ﬁeld Q √ l . Since l is square free, the ring= of− integers of K is − 1 √ l a b√ l Z " + − # ( + − a, b Z, a b (mod 2)) . O = 2 = 2 : ∈ ≡

In particular is a Dedekind domain. We shall show that some of the familiar lattices L in l 1 O R + are modules for the ring , that is that αL L whenever α . Let O ⊆ ∈ O n L0 (a1,..., an) Z a1 an 0 (mod 2) = { ∈ : + · · · + ≡ } be the Dn root lattice.

LEMMA 3.1. The lattice L0 is an -module. O 1 1 PROOF. It sufﬁces to show that 1 √ l v v(I W) L0 whenever v L0. Indeed 2 + − = 2 + ∈ ∈ it sufﬁces to show this whenever v lies in a generating set for L0. Now L0 is generated by the vectors 2e j and e j ek (for j k) where e j denotes the jth unit vector. Firstly e j (I W) is a row of the Hadamard+ matrix6=I W. As it contains n instances of 1 and n is even,+ it lies 1 + ± in L0. Next 2 (e j ek)(I W) is the sum of two rows of the Hadamard matrix I W. Two + + 1 + rows of an n-by-n Hadamard matrix agree in exactly n/2 places. Hence 2 (e j ek)(I W) 1 + + has n/2 zeros and n/2 instances of 1. As n/2 is even then 2 (e j ek)(I W) L0. This completes the proof. ± + + ∈ 2 Unimodular lattices 1035

Now consider the set

(a1,..., an) a j 1/2, 1/2 . S = { : ∈ {− }} The difference of two vectors in lies in L0 if and only if those vectors agree in an even S number of places. Thus there are exactly two cosets v L0 as v runs through . The union of with either of these cosets is a lattice and it will be+ convenient to distinguishS these lattices byS a notation dependent on the matrix W. 1 1 For each j, 2 e j (I W) , and for each j and k, 2 (e j ek)(I W) L0 by Lemma 3.1. 1 + ∈ S − + ∈ Thus the cosets e j (I W) L0 are identical. Let 2 + + 1 v v e1(I W) L0 S+ = { ∈ S : − 2 + ∈ } and . S− = S \ S+ 1 1 1 As e j (I W) e j ( I W) e j / L0 then e j ( I W) for each j. 2 + − 2 − + = ∈ 2 − + ∈ S− If v then 2v has n entries 1 and so 2v L0. It follows that L0 (v L0) is a lattice, ∈ S ± ∈ ∪ + which depends only on whether v or v . We write L for L0 (v L0) when ∈ S+ ∈ S− + ∪ + v and L for L0 (v L0) when v . Both L and L are isometric to the lattice ∈ S+ − ∪ + ∈ S− + − usually denoted by Dn+ [5, Chapter 4, Section 7.3]. The lattice Dn+ is unimodular for each n divisible by 4, and it is even unimodular whenever n is divisible by 8.

LEMMA 3.2. If n is divisible by 8 then the lattices L and L are -modules. + − O

PROOF. Let L L or L . Then L L0 (v L0) for some v and by Lemma 3.1 = +1 − =1 + + 1 ∈ S it suffices to show that 2 1 √ l v 2 v(I W) L. Note that 4 (l 1) is an even integer by the hypothesis. + − = + ∈ + 1 1 We may assume that v e1( I W). If v e1(I W) then = 2 ± + = 2 + 1 1 2 1 1 l 1 v(I W) e1(I W) e1((1 l)I 2W) e1(I W) + e1 2 + = 4 + = 4 − + = 2 + − 4 1 which lies in L as 2 e1(I W) L. 1 + ∈ If v e1( I W) then = 2 − + 1 1 1 v(I W) e1( I W)(I W) e1( (l 1)I ) 2 + = 4 − + + = 4 − + which lies in L. 2 Let be an ideal of . If M is a -module, then M, defined as the subgroup of M generatedI by the αm for Oα and m OM, is also a -module.I Recall that the norm N( ) of the ideal of is the index∈ I . Each∈ ideal of Ois of the form α, β α β .I The ideal classesI ofO K are the equivalence|O : I| classes of nonzeroO ideals of under = theO equivalence+ O relation if ξ where ξ Q √ l . There are finitely manyO ideal classes of K ; their numberI ∼ J is theIclass-number= J of ∈K . (See − [3] for the theory of ideal classes in quadratic fields.)

THEOREM 3.1. Suppose that l 7 (mod 8) and that is a nonzero ideal of with norm N N( ). If L L or L then≡ I O = I = + − 1 L L [I] = √N I is an even unimodular lattice. Also if and lie in the same ideal class of , the lattices L and L are isometric. I J O [I] [J ] 1036 R. Chapman

PROOF. First of all we show that the index L L equals N n/2. As an -module, L is finitely generated. Also if α and v |L are: I nonzero,| then αv αO v 0 by Lemma 2.1 and so L is torsion free∈ O as an -module.∈ | | = | || | 6= By the theory of modules over DedekindO domains [4, Section 9.6], as L is a finitely generated torsion-free module over the Dedekind domain , then L L1 Lk where each L j O = ⊕· · ·⊕ is isomorphic to a nonzero ideal j of . Each of the j is a free Abelian group of rank 2, and A O A as L is a free Abelian group of rank n it follows that k n/2. Then L L1 Ln/2 n/2 = I = I ⊕ · · · ⊕ I and so L L j 1 L j L j . But | : I | = Q = | : I | j N( j ) L j L j j j |O : IA | IA . | : I | = |A : IA | = j = N( j ) |O : A | A But N( j ) N( )N( j ) and so L j L j N( ) N. Consequently L L N n/2 asIA claimed.= I A | : I | = I = | : I | = We now show that L is unimodular. The lattice L is generated by elements u αv [I] I = where α and v L. Let u j α j v j ( j 1, 2) with α j and v j L. Then by Lemma 2.1,∈ I ∈ = = ∈ I ∈ u1 u2 (α1v1) (α2v2) (α1α∗v1) v2. · = · = 2 · But α1α N( ) [3, Section VIII.1] so that 2∗ ∈ II∗ = I O u1 u2 N(γ )v1 v2 · = · where γ . As γ v1 L (by Lemma 3.2) and L is an integral lattice, then ui u2 0 ∈ O ∈ 1/2 · ≡ (mod N). Consequently L N − L is an integral lattice. But L is unimodular, so it has [I] = I n/2 1/2 volume 1. Thus L has volume L L N and so L N − L has volume 1. Thus L is a unimodularI lattice.| : I | = [I] = I We finally[I] show that L is an even unimodular lattice. Since L is integral, to show that it is even it suffices to show[I] that each vector u in a generating set[ ofI]L has u 2 even. The vectors u N 1/2αv for α and v L generate L . Then [I] | | = − ∈ I ∈ [I] 1 α 2 u 2 αv 2 | | v 2. | | = N | | = N | | 2 2 2 But α αα∗ ∗ N and so α /N Q Z and v is an even integer, as v |L, an| = even lattice.∈ II Thus= uO2 is an even| | integer.∈ Thus∩ OL= is an| even| unimodular lattice. ∈Now suppose that and | |lie in the same ideal class of[I] . Then α where α is a I J O J = I nonzero element of K . Then L α L and so J = J 1 1 N( ) L L α L s I αL . [J ] = √N( )J = √N( ) I = N( ) [I] J J J Let γ α√N( )/N( ). Since α then N( ) α 2 N( ) and so γ 1. By Lemma= 2.1, the mapI v J γ v is anJ isometry= I of Rn andJ as=L | | γI L , the| lattices| = L and L are isometric.7→ [J ] = [I] [I2] [J ] Given L, we can produce a maximum of h nonisometric lattices L where h denotes the class-number of the quadratic field K . [I] It is useful to note which for which ideals is L Zn. I I + ⊆ n 1 LEMMA 3.3. Let be an ideal of . Then L Z if and only if 2, 2 1 √ l . In this case also N( I)Zn L . O I + ⊆ I ⊆ − − I ⊆ I + Unimodular lattices 1037

n n PROOF. Note that L Z L0 and so L Z if and only if L L0. This occurs + ∩ = I + ⊆ I + ⊆ if and only if annihilates the -module M L /L0. This module has two elements, I O = + 1 so it must be isomorphic to / where is an ideal of norm 2. As 2, 2 1 √ l has O J 1 J 1 1 − − norm 2 and is seen to annihilate M as 2 1 √ l takes 2 e1(I W) to 4 (l 1)e1, then 1 − − + + 2, 2 1 √ l . Thus is the annihilator of M and the first statement follows. J = − − 1 J Suppose that 2, 2 1 √ l . The lattice L is unimodular so that if u v Z for all v L I ⊆then u −L − . If u √N( )+w[Iwith] w Zn then u v Z· for∈ all ∈ 1/2+[nI] ∈ +[I] 1=/2 n I ∈ · ∈n v N( )− Z and as L N( )− Z then u L . Hence √N( )Z L and∈ so NI ( )Zn L . +[I] ⊆ I ∈ +[I] I ⊆ +[I2] I ⊆ I + In this case the lattice 3 is the inverse image of a subgroup of (Z/NZ)n, where N N( ), under the projection π Zn (Z/NZ)n. Such a subgroupC is called a linear code of= I : → length n over Z/NZ. We also say that 3 is obtained from by construction AN . The standard dot product is well defined on the group (Z/CNZ)n. If a subgroup (Z/NZ)n satisfies u v 0 for all u, v then is self-orthogonal. Also is self-dual Cif ⊆u 0 if and only if· u = . By the nondegeneracy∈ C C of the dot product, isC self-dual if and only· C = if is self-orthogonal∈ andC N n/2. C C |C| = PROPOSITION 3.1. Let be an ideal of with 2, 1 1 √ l and N( ) N. The I O I ⊆ 2 − − I = lattice L L is obtained from construction AN from a self-dual linear code of length n over Z/N=Z.I + C 1 2 If N, 2 a √ l with a 1 (mod 4) and a l (mod 4N) then is spanned by I = − −1 ≡ ≡ − C the vectors of the form (ei e j )(aI W) (1 i j n). 2 + − ≤ ≤ ≤ PROOF. Apart from the self-duality of we have already proved the first assertion. The C 1/2 self-duality of follows from the unimodularity of N − L . By volume considerations C I + N n/2 Zn L (Z/NZ)n = | : I +| = | : C| n/2 1/2 1/2 and so N . Also if u, v L then N − u and N − v lie in the integral lattice 1/2 |C| = 1 ∈ I + N − L so that N − u v Z. Hence is self-orthogonal, and as it has the correct order, it is self-dual.I + · ∈ C 1 The ideal contains the subgroup NZ 2 a √ l Z of and as this subgroup also has I 1 + − − O 1 index N in then NZ 2 a √ l Z. It follows that L NL 2 a √ l O I = 1 + − − 1 I + = + + − − L . As a 1 (mod 4), 2 a √ l 2 1 √ l is an even integer. It follows that + 1 ≡ 1 + − − + − 1 L0 2 e1(aI W) L0 2 e1(I W) and so L is generated by L0 and u 2 e1(aI W). + + = + + + 1 = + Thus NL is generated by the N(ei e j ) and Nu and 2 a √ l L is generated by the 1 + + − − + (ei e j )(aI W) and 2 + − 2 1 1 a l u(aI W) e1(aI W)(aI W) + e1. 2 − = 4 − + = 4 2 1 Note that (a l)/4 is a multiple of N. It follows that is generated by Nu and the (ei e j ) + C 2 + (aI W). But Nu (N/2)e1(I W) is congruent modulo N to the word consisting of all − = + N/2s. Also (N/2)e1(aI W) is congruent to the same word. We can drop the generator Nu − 1 and deduce that is generated by the (ei e j )(aI W). 2 C 2 + −

4. QUADRATIC RESIDUE CODES

To use the above construction of lattices, we need a supply of skew-symmetric conference matrices. Paley [8] constructed a family of such matrices of order n l 1 whenever l 3 = + ≡ 1038 R. Chapman

(mod 4) is prime. To apply our theory we stipulate in addition that l 7 (mod 8). We ﬁnd that the lattices L are derived from quadratic residue codes in this case.≡ I + We deﬁne a conference matrix W n as follows. Let ∈ W 0 1 1  1 ···  W −. =  .   . W 0   1  − where the l-by-l matrix W 0 is the circulant matrix whose (i, j)-entry is j i W 0 − i j = l where ( ) denotes the Legendre symbol. This matrix W is called a conference matrix of Paley type. For the rest of this section W will denote this particular matrix. We follow the usual practice with quadratic residue codes and label the entries of a typi- cal vector of length n l 1 using the elements of the projective line over Fl as follows: = + v (v , v0, v1, v2, . . . , vl 1). We let e , e0, e1,..., el 1 denote the corresponding unit vec- = ∞ − ∞ − tors, that is, eµ has a one in the position labelled µ, and zeros elsewhere. LEMMA 4.1 (PALEY). The matrix W is a skew-symmetric conference matrix.

PROOF. See for instance [7, Chapter 18]. 2 In , the ideal 2 splits as a product of two distinct prime ideals: 2 where 1 O O 1 O = PQ P =r 2, 2 1 √ l and ∗ 2, 2 1 √ l . We shall investigate the lattices L and L + r −for integersQ =r P0.= (The discussion − − for L r and L r is similar.) +[P ] We+ ﬁrst[Qneed] a lemma on≥ the structure of the ideals−[Pr and] r . −[Q ] P Q LEMMA 4.2. Let r be a positive integer. Then r 2r Z 1 t √ l Z P = + 2 + − and r 2r Z 1 t √ l Z Q = + 2 − − where t is any integer with t2 l (mod 2r 2) and t 1 (mod 4). ≡ − + ≡ PROOF. It is well known that if s 3, and a 1 (mod 8) then the congruence x2 a s ≥ 2 ≡ r 2 ≡ (mod 2 ) is soluble. Thus there exists t with t l (mod 2 + ). By replacing t by t if ≡ − r 1 − necessary, we may assume that t 1 (mod 4). Consider the ideal 2 , 2 t √ l of . r r ≡ r r 1 I 1= + − O As 2 then is a factor of 2 . But as 2 t √ l 2 1 √ l 2(t 1)/4 ∈ I I 1O = P Q + − = + − + − ∈ then is a factor of . But t √ l / , and so 2 K is not a factor of . P P I 4 + − ∈ O O = PQ I Hence r0 where 1 r r. Letting α 1 t √ l we have I = P ≤ 0 ≤ = 2 + − r r ∗ 2 , α 2 , α∗

II = 2r r r 2 , 2 α, 2 α∗, αα∗

= 2r r r 2 2 , 2 α, 2 α∗,(t l)/4 = + 2r ⊆ O 2 r 2 r r as t l (mod 2 + ). But ∗ N( ) 2 0 and so r r 0, that is . ≡ −r r 1 II = I O = O = r I = P Now 2 Z 2 t √ l Z, but both these groups have index 2 in so they are equal. The statementP ⊆ about+ r +now− follows by complex conjugation. O 2 Q Unimodular lattices 1039

r r 1 We now consider the lattices L for r 1. Since and 2, 2 1 √ l r Q + ≥ Q ⊆ Q Q = − − then by Proposition 3.1 L is obtained by construction A2r from a self-dual code r over r Q + C Z/2 Z. We shall show that r is the Hensel lift of an extended quadratic residue code in the sense of [1]. C 2 r 2 Recall that the integer t satisfies t 1 (mod 4) and t l (mod 2 + ). By Proposition 3.1 ≡ 1 ≡ − it follows that r is generated by the vectors 2 (ei e j )(W t I ). It is plain that we need only C + − 1 those vectors with i and so r is spanned by u e (W t I ) and v j 2 (e e j ) (W t I ) for 0 j <=l ∞. C = ∞ − = ∞ + Let− φ (Z/2≤r Z)n (Z/2r Z)l be the map given by deleting the first coordinate of the : → vector. The code r contains the vector u ( t, 1, 1,..., 1). As r is odd and r is self-dual, C = − C the intersection of r and the kernel of φ is trivial. Thus φ( r ) has the same order C Cr0 = C as r . Then φ(u) is the all-ones vector, and φ(v j ) are cyclic shifts of φ(v0). Also φ(v0) C = (c0, c1,..., cp 1) where − (1 t)/2 if j 0, − = c j ( 1 if j is a quadratic residue modulo l, = 0 if j is a quadratic nonresidue modulo l. r Thus r0 is a cyclic code over Z/2 Z. WeC recall the definition of quadratic residue codes. Consider the polynomial Xl 1 over l − the field F2 Z/2Z. Then X 1 splits into linear factors in some finite extension F2k of F2. In fact = − l 1 − Xl 1 (X ζ j ) − = Y − j 0 = where ζ is a primitive lth root of unity in F2k . We write Xl 1 (X 1) f (X) f (X) − = − + − where f (X) (X ζ j ) and f (X) (X ζ j ). + = Y − − = Y − ( j/l) 1 ( j/l) 1 = =− As l 7 (mod 8), then 2 is a quadratic residue modulo l, and so the coefficients of both f ≡ 2 + and f are invariant under the Frobenius automorphism δ δ of F2k . Consequently both − 7→ f and f have coefficients in F2. The labelling of these factors as f and f depends on the choice+ of−ζ. Replacing ζ by another primitive lth root of unity either preserves+ − or interchanges (l 3)/2 f and f . The coefficients of X − in f and f are 0 and 1 is some order, so we can, and+ shall,− choose ζ such that + − (l 1)/2 (l 3)/2 (l 1)/2 (l 3)/2 f (X) X − 0X − and f (X) X − X − . + = + + · · · − = + + · · · The cyclic codes of length l over F2 with generator polynomials f (X) and f (X) are called the quadratic residue codes. + − Bonnecaze, Sole´ and Calderbank [1] extended the notion of quadratic residue code to codes over Z/2r Z. By Hensel’s lemma there exist unique polynomials f (r)(X) and f (r)(X) with coefficients in Z/2r Z such that + − Xl 1 (X 1) f (r)(X) f (r)(X), − = − + − f (r)(X) f (X)(mod 2) and f (r)(X) f (X)(mod 2). + ≡ + − ≡ − The cyclic codes over Zl with generator polynomials f (r)(X) and f (r)(X) are called lifted quadratic residue codes over Z/2r Z. + − 1040 R. Chapman

r THEOREM 4.1. The code r0 is the lifted quadratic residue code over Z/2 Z with generator (r) C polynomial f (X). + PROOF. Cyclic codes of length l over R Z/2r Z correspond to ideals of the polynomial ring R X / Xl 1 . The code corresponds= to the ideal g, h where [ ] − Cr0 I = p 1 − g(X) X j = X j 0 = and 1 t h(X) − X j . = 2 + X ( j/l) 1 = We ﬁrst consider the case where r 1. Then u(X) where u(X) is the greatest = I = common divisor of g(X) and h(X). Let ζ be a root of f (X) 0 in an extension ﬁeld of F2. j - + = 1 The roots of g(X) are the ζ where p j. As t 1 (mod 4) then 2 (1 t) is even and so j ≡ − h(X) ( j/l) 1 X . Now = P = ζ j 0 and ζ j 1. X = X = ( j/l) 1 ( j/l) 1 = =− It follows that h(ζ a) (ζ a) j 0 = X = ( j/l) 1 = a if and only if l 1. Thus u(X) f (X). Now we consider = the general case.= The+ reduction of modulo 2 is . Any liftings to Cr0 C10 Cr0 of a basis of generate a free R-module (of rank 1 (l 1)), and so they generate the whole C10 2 + code r0 . As r is free over R, it is generated as an ideal by a monic polynomial F(X), of degreeC 1 (l C1). As F(X) reduces to f (X) modulo 2, and F(X) Xl 1 it follows that 2 + + | − F(X) f (r)(X) as required. 2 = + Given the code , the code r can be reconstructed, since for each element of the corre- Cr0 C Cr0 sponding element of r is uniquely determined as it is orthogonal to ( t, 1, 1,..., 1). We now turn to rCL . This is no longer a sublattice of Zn. − P + LEMMA 4.3. Let r be a positive integer. The index r L r L Zn 2. The lattice r n r |P + : P + ∩ | = L Z is generated by the vectors 2 (e eµ) (µ , 0, 1, 2,..., l 1 ), the vector P + ∩ ∞1 + ∈ {∞ − } r u e (W t I ) and and the vectors v j (e e j )(W t I ) (0 j < l). Also L is = ∞ + = 2 ∞ + + ≤ P + generated by r L Zn and 1 u 1 (t2 l)e . P + ∩ 2 − 4 + ∞ r r 1 PROOF. We have 2 Z 2 t √ l Z. Let 0 denote the lattice generated by the r P = + + − 1 2 (e eµ), u and the v j . The lattice L is generated by the e eµ and 2 e (t I W). ∞ r+ 1 + 1 ∞ + ∞ +r Thus 2 (e eµ), u 2 t √ l 2e and v j 2 t √ l (e e j ) all lie in L . ∞ + = + − ∞ = +r − n∞ + P + These vectors all have integer coordinates, and so 0 L Z . 1 1 2⊆ P + ∩ Let be the lattice generated by 0 and u (t l)e . Now 2 − 4 + ∞ 1 t √ l 1 e (t I W) 1 t √ l 2e 2 + − 2 ∞ + = 4 + − ∞ t t2 l " t √ l + # e = 2 + − − 4 ∞ t t2 l u + e . = 2 − 4 ∞ Unimodular lattices 1041

r As t is odd and u 0 then L . r ∈ ⊆ P + r 1 r 1 1 2 The lattice L is generated by and 2 − e (t I W) 2 − u. But u 2 (t l)e 2 P + r 1 ∞ + = r 1 − 1 +2 ∞ ∈ and as t l is divisible by 2 + then u 0 and so L . As 2 u 4 (t l)e is n+ ∈ =r P + r −n + ∞ not in Z but its double is in 0, then 0 L L Z 2 and so r n | : | = |P + : P + ∩ | = 0 L Z . 2 = P + ∩ One can now proceed to express the lattices r L and r L Zn in terms of lifted quadratic residue codes over Z/2r Z. For simplicityP we+ presentP the+ ∩ details only for r 1. = Let 0 denote the cyclic quadratic residue code of length l over F2 with generator polynomial f (DX), and let denote its extension obtained by appending a parity check bit at the front (so− that the entriesD in each word of sum to zero). D

THEOREM 4.2. The lattice L Zn consists of those vectors reducing modulo 2 to elements of and the sum of whoseP entries+ ∩ is a multiple of 4. The lattice L is obtained from L ZDn by adjoining the extra generator 1 ( 1 (1 l), 1, 1,..., 1). P + P + ∩ 2 2 −

PROOF. We may take t 1 in the proof of Lemma 4.3. In this case the vector u is the = 1 1 1 all-ones vector while each v j consists of 2 (l 1) ones and 2 (l 1) zeros. As 2 (l 1) is a multiple of 4 the sum of the entries of each+ of these vectors is+ a multiple of 4. As+ this is manifestly true for the vectors 2(e eµ) too, then the sum of the entries of each vector in L Zn is a multiple of 4. ∞ + P If+ we∩ delete the ﬁrst entry of the given generators of L and reduce modulo 2 we get the P + all-ones vector of length l and the cyclic shifts of the vector w0 (d0, d1,..., dl 1) where = − 1 if j 0 or if j is a quadratic residue modulo l, d j = = 0 if j is a quadratic nonresidue modulo l.

By a similar argument to the proof of Theorem 4.1 these vectors generate the cyclic quadratic n residue code 0. Hence each element of L Z reduces modulo 2 to an element of . If denotes theD sublattice of Zn consistingP of+ vectors∩ reducing modulo 2 to and with theD n 1 n/2 n D n entries summing to a multiple of 4, then Z 2 + Z L Z . Thus L Zn. | : | = = | : P + ∩ | = P + ∩ 1 1 2 1 1 Now letting t 1 we see that 2 u 4 (t l)e 2 ( 2 (1 l), 1, 1,..., 1) and so this vector together with= L Zn generates− L+. ∞ = − 2 P + ∩ P + In the terminology of Conway and Sloane [5, Chapter 5, Section 3], the lattice L Zn is obtained from the code by construction B. Then the lattice L is obtainedP by+ density∩ doubling. One can extend theseD notions to lifted quadratic residueP codes+ to produce the lattices r L . P We+ look brieﬂy at the lattices L for more general ideals . Consider the case where , an ideal of norm p, an oddI prime.+ Then p, t √ Il where t2 l (mod p). I = A A = + − ≡ − The rows of the matrix t I W generate a self-dual linear code over Fp which turns out to be an extended quadratic residue+ code. The coordinates of vectorsC in L are half-integers, and it is meaningful to reduce these modulo the odd prime p. Then the lattice+ L simply consists of the vectors in L which reduce modulo p to elements of . MoreA generally+ r L will have a similar description+ in terms of an extended lifted quadraticC residue code overA Z+/pr Z. Finally by splitting a general ideal into a product of powers of prime ideals r , we can describe L in terms of the variousI r using the Chinese remainder theorem. A I + A 1042 R. Chapman

5. EXAMPLES

Since the ring Z 1 1 √ 7 has class number 1 (and each even unimodular rank 8 lattice [ 2 + − ] is isometric to the E8 root lattice) the first interesting examples occur when l 15 and the first interesting examples with Paley matrices occur when l 23. = = 5.1. l 23 and l 31. In both these cases we take W to be the Paley matrix. We first consider= the case l =23. = 1 The class group of Z 2 1 √ 23 has order 3, and the class of each of its ideals 1 O = + 1 − 2, 2 1 √ 23 and 2, 2 1 √ 23 generates its class group. The lattice L P = + − Q = r − − + itself is the lattice D+ . The lattices L are obtained by applying construction A2r to the 24 Q + lifted quadratic residue codes r . The code 1 is the extended binary Golay code. It is plain that L is obtained by applyingC constructionC A [Chapter 5, Section 2] to the binary Golay Q + 24 code, and so L is isometric to the Niemeier lattice with root system A1 . The isometry+[ classesQ] of the unimodular lattices L r depend only on the congruence +r [Q ] class of r modulo 3. If r 0 (mod 3) then L is isometric to D24+ while if r 1 r ≡ +[Q ] 24 ≡ (mod 3) then L is isometric to the Niemeier lattice with root system A1 . To identify L r when r+[Q2] (mod 3) note that 2 lies in the same ideal class as . Hence for r 2 (mod+[Q 3),] L r ≡is isometric to L Q. By Theorem 4.2 it is plain that PL is the Leech≡ lattice, as given+[Q by] Leech’s original+ construction[P] [6]. Applying Theorem 3.1+[ givesP] an explicit isomorphism between L and L 2 which is equivalent to that constructed in [2]. In general if s is the+ order[P] of the+[ classQ ] of the ideal in the class group of , then up to isometry L r and L r depend only on the congruenceP class of r moduloO s. Also r +[P r] +[Q ] L and L 0 will be isometric whenever r r 0 (mod s). For l 31 we also have s +[P3 and] the+ above[Q ] discussion is valid for l 31≡ too. − In particular L = is isometric to L = 2 , and we recover [2, Theorem 1]. = +[P] +We[Q can] give alternative constructions of the Leech lattice at will simply by writing down ideals of Z 1 1 √ 23 equivalent to . Let 3, 1 1 √ 23 and 3, 1 1 2 + − P I = 2 + − J = 2 − + √ 23 . Then , and 6, 1 5 √ 23 all lie in the same ideal class. − P J QI = 2 − + − The lattice L is generated using density doubling from the lattice L0 consisting of all vectors in Z24Jwhose+ entries sum to zero and which reduce modulo 3 to elements of the extended ternary quadratic residue code with generator matrix I W. Then L is generated 1 −1/2 J + by L0 and the vector 2 (5, 1, 1,..., 1). The lattice L 3− L is isometric to the Leech lattice. +[J ] = J + Next consider the lattice L . This consists of the vectors in Z24 reducing modulo 2 and modulo 3 to elements of appropriatelyQI + chosen binary and ternary quadratic residue codes. 1 The binary code in question is that generated by vectors 2 (e eα)(I W) for α , 0, 1, 2,..., l 1 and the ternary code is generated by the rows∞of+I W. Then− L ∈ {∞1/2 − } + +[QI] = 6− L is isometric to the Leech lattice. QI + 5.2. l 47. Again we take W to be the Paley matrix. In [5, Chapter 7, Section 7] the lattice = 3 P48q is described. This is an even unimodular lattice of rank 48 and minimum norm 6. = It is generated by the following vectors (a , a0, a1,..., a46): ∞ (i) 1/√12 ( 5, 1, 1,..., 1), (ii) those vectors − of the shape 1/√3 (124, 024) supported on the translates modulo 47 of the set 0 Q where Q is the set of quadratic residues modulo 47, (iii) all vectors{ } ∪ of the shape 1/√3 ( 32, 046). ± It is more convenient to consider instead the equivalent lattice 30 generated by the vectors Unimodular lattices 1043

(i) 1/√12 (5, 1, 1,..., 1), 0 24 24 (ii)0 those vectors of the shape 1/√3 (1 , 0 ) supported on the translates modulo 47 of the set 0 N where N is the set of quadratic nonresidues modulo 47, { } ∪ (iii) all vectors of the shape 1/√3 ( 32, 046). 0 ± 1 We claim that 30 is the lattice L where 3, 1 √ 47 . Note that the norm of +[I] I = 2 − − I is 3. It suffices to show that each of the generating vectors for 30 is contained in L . Since each vector of shape ( 12, 046) lies in L and 3 P then it is immediate that+ the[I] vectors ± + ∈ of type (iii)0 lie in L . The vectors of type (ii)0 are the differences of the first row and an +[I] √ 1 arbitrary other row of the matrix 1/2 3 (I W). Since 2 1 √ 47 , the vectors of 1 − − −1 ∈ I type (ii)0 lie in L . Finally, 2 (1, 1, 1,..., 1), the first row of 2 (I W), lies in L . 1 +[I] − −1 − − I + Also v0 2 e0(I W) L and 3v0 2 (3, 3, 3,..., 3) L . Adding these two vectors 1 = + ∈ + = ∈ I + gives 2 (5, 1, 1,..., 1) L so that the vector of type (i)0 does lie in L . 1 ∈ I + 2 +[I] The ideal 2 1 √ 47 has norm 12 and factors as . The class number of Q √ 47 − 2− 3 Q I 3 − is 5, and so . Thus 30 is isometric to L , which is constructed using construction[I A] from = [P the] = quadratic [Q ] residue code of length 48+[ overQ ] Z/8Z.

5.3. l 15. In this case there is no Paley matrix. We consider two different conference matrices= of order 16. If W n then the 2n-by-2n matrix ∈ W WI W W 0 + = I W W − + − is a skew-symmetric conference matrix of order 2n. Applying this construction four times to the zero matrix in 1 gives the matrix W 0  +++++++++++++++0  − −+−+−+−+−+−+−+  0   − + −−++−−++−−++−   0   − − + −−++−−++−−++   0   − + + + −−−−++++−−−   0   − − − + + + − − − − + + + + −   0   − + − − + − + − + − − + − + +   0  W1  − − + − + + − − − + − + + − +  =  0   − + + + + + + + − − − − − − −   0   − − − + − + − + + + − + − + −   0   − + − − − + + − + − + + − − +   0   −−+−−−++++− + + − −   0   −+++−−−−+−−− + + +   0   −−−++−+−+++−− − +   0   −+−−+−−++−++−+ −0  −−+−++−−++−+−−+ where, for convenience, we have denoted 1 and 1 by and respectively. The ideal class group of Z 1 1 √ 15 has order 2. The ideal− +2, 1 1 − √ 15 is not principal and 2 + − I = 2 − − 1044 R. Chapman

L is given by construction A from the binary code with the generator matrix I + 10010101000000 0 0  0101001100000000   0011011000000000     0000111100000000    .  0000000010000111     0000000001001011     0000000000101101   0000000000011110 

Thus L is isometric to the orthogonal direct sum of two copies of the D8+ lattice. This is not isometric+[I] to L . Another conference+ matrix of order 16 is 0  +++++++++++++++0  − ++−+−−+−++−+−−  0   − − ++−+−+−−++−+−   0   − − − ++−++−−−++−+   0   − + − − ++−++−−−++−   0   − − + − − + + + − + − − − + +   0   − + − + − − + + + − + − − − +   0  W2  − + + − + − − + + + − + − − −  . =  0   − − − − − − − − + + + + + + +   0   − + + + − + − − − − − + − + +   0   − − + + + − + − − + − − + − +   0   −−−+++−+−++ − − + −   0   −+−−+++−−−++ − − +   0   −−+−−+++−+−++ − −   0   −+−+−−++−−+−++ −0  −++−+−−+−−−+−++ In this case the lattice L is obtained using construction A applied to the binary code with generator matrix I + 1000000001111111  0100000010111111   0010000011011111     0001000011101111    .  0000100011110111     0000010011111011     0000001011111101   0000000111111110 

Thus L is isometric to the D16+ lattice and so to L . This example shows that the isometry class of+L[I] depends on the choice of the conference+ matrix W, and also that L and L may+[I be] isometric even when and are in different ideal classes. +[I] +[J ] I J

REFERENCES

1. A. Bonnecaze, P. Sole´ and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inf. Theory, 41 (1995), 366–377. Unimodular lattices 1045

2. R. Chapman and P. Sole,´ Universal codes and unimodular lattices, J. Theor.´ Nombres Bordeaux, 8 (1996), 369–376. 3. H. Cohn, A Second Course in Number Theory, Wiley, 1962. 4. P. M. Cohn, Algebra, 2nd edn, Wiley, 1989, Vol. 2. 5. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 1988. 6. J. Leech, Notes on sphere packing, Can. J. Math., 19 (1967), 251–267. 7. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992. 8. R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys., 12 (1933), 311–320. 9. J. -P. Serre, A Course in Arithmetic, Springer, 1973.

Received 21 July 2000 and accepted 26 June 2001 in revised form 20 June 2001

ROBIN CHAPMAN School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, U.K. E-mail: [email protected]