Linear Codes and Geometry Over Finite Chain Rings

Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Linear Codes and Geometry over Finite Motivation

Finite Chain Chain Rings Rings

Modules over Finite Chain Rings Thomas Honold Linear Codes over Finite Chain Rings Institute of Information and Communication Engineering Projective and Zhejiang University Afﬁne Hjelmslev Spaces ZiF Cooperation Group on Finite Projective Ring Linear Code Constructions Geometries Fine Structure August 2009 Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Outline

Thomas Honold 1 Motivation

Motivation 2 Finite Chain Rings Finite Chain Rings 3 Modules over Finite Chain Rings Modules over Finite Chain Rings 4 Linear Codes over Finite Chain Rings Linear Codes over Finite 5 Projective and Afﬁne Hjelmslev Spaces Chain Rings

Projective and Afﬁne 6 Linear Code Constructions Hjelmslev Spaces 7 Fine Structure Linear Code Constructions 8 Singer’s Theorem Fine Structure

Singer’s Theorem 9 Arcs and Blocking Sets

Arcs and Blocking Sets 10 Hyperovals and Ovals

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Rings

Modules over Finite Chain Today’s Lecture: Linear codes and projective Rings Hjelmslev geometry over ﬁnite chain rings Linear Codes over Finite Chain Rings

Projective and Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Speaker Rings Dr. Thomas Honold (Assoc. Prof.) Modules over Finite Chain Dept. of Information Science and Electronics Engineering Rings Zhejiang University, Zheda Road Linear Codes over Finite Email: [email protected] Chain Rings

Projective and Afﬁne The talk describes joint work with IVAN LANDJEV,MICHAEL Hjelmslev Spaces KIERMAIER,AXEL KOHNERT,SILVIA BOUMOVA,...

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Disclaimer

Thomas Honold

Motivation

Finite Chain Rings In what follows . . . Modules over Finite Chain • rings will be associative with identity 1 6= 0 (no rngs, Rings please!) Linear Codes over Finite Chain Rings • modules will be unital (no modls), ring homomorphisms

Projective and will preserve 1 (no homomorhisms) Affine Hjelmslev • “finite” means “of finite cardinality” (not only “finitely Spaces generated”) Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Algebraic Coding Theory Suppose A is a ﬁnite set of size |A| = q ≥ 2 (“q-ary alphabet”). Thomas Honold Deﬁnition n Motivation A (block) code over A is a nonempty subset C ⊆ A for some

Finite Chain integer n ≥ 0. Rings

Modules over Parameters of a code Finite Chain Rings n Length, i.e. C ⊆ An Linear Codes over Finite M Number of codewords, i.e. M = |C| Chain Rings Projective and d Minimum distance, deﬁned as Afﬁne ′ ′ ′ Hjelmslev min{dHam(c, c ); c, c ∈ C, c 6= c } Spaces Linear Code A code with these parameters is said to be a q-ary (n, M, d) code Constructions or, using bracket notation, an [n, k, d] code, where k = log M. Fine Structure q

Singer’s The performance of a communication system using C to transmit Theorem information over a noisy channel is determined by k/n Arcs and Blocking Sets (transmission rate) and d/n (error correction rate), which both

Hyperovals should be large. and Ovals For complexity reasons n should be small. Linear Codes and Geometry over Finite Chain Rings Main Problem

Thomas Honold

Motivation

Finite Chain Rings

Modules over Finite Chain Rings • Given two of the parameters (n, M, d), optimize the Linear Codes value of the third parameter. over Finite Chain Rings • Find codes with sufﬁciently rich structure in order to Projective and Afﬁne facilitate (encoding) and decoding. Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Classical Linear Codes Suppose q > 1 is a prime power and A = Fq = GF(q) is the ﬁnite Thomas Honold ﬁeld of order q.

Motivation Deﬁnition

Finite Chain A q-ary linear [n, k, d] code is a code over Fq with parameters Rings k n [n, k, d] (or (n, 2 , d)), which is also a vector subspace of Fq. Modules over Finite Chain Rings Simpliﬁcations Linear Codes • k = dim(C) over Finite Chain Rings • d = min wHam(c); c ∈ C; c 6= 0 , where wHam(c) is the Projective and Afﬁne Hamming weight (number of nonzero entries) of c Hjelmslev Spaces k k×n • C = {xG; x ∈ Fq}, where G ∈ Fq is a matrix having as its Linear Code rows a basis of C (generator matrix). Constructions

Fine Structure n T (n−k)×n • C = {c ∈ Fq; Hc = 0} for some H ∈ Fq (parity-check Singer’s matrix) Theorem Arcs and For some families of linear codes (Reed-Muller codes, cyclic Blocking Sets codes, algebraic-geometry codes, low-density parity-check Hyperovals and Ovals codes) efﬁcient encoding and decoding algorithms are known. Linear Codes and Geometry over Finite Chain Rings The Hexacode

Thomas A particularly nice example Honold

Motivation Finite Chain The Hexacode is the quaternary linear code with generator matrix Rings

Modules over Finite Chain 100111 Rings G = 0 1 0 1 α β ∈ F3×6,   4 Linear Codes 0 0 1 1 β α over Finite Chain Rings   2 Projective and where F4 = {0, 1,α,β} (so β = 1 + α = α ). Afﬁne Hjelmslev C has parameters n = 6, k = 3 (so |C| = 23 = 8), d = 4. Spaces Linear Code C is an MDS code, i.e. it satisﬁes the Singleton bound Constructions d n k 1 with equality. In terms of G this means that every Fine Structure ≤ − +

Singer’s 3 × 3-submatrix of G is invertible. Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Geometric View

Thomas Idea: View the columns of G as (homogeneous coordinates) of Honold points of PG(2, F4).

Motivation The projective plane over F4 Finite Chain Rings PG(2, F4) is deﬁned as the point-line incidence structure (P, L, I)

Modules over with Finite Chain Rings 3 3 P := {U ≤ F4; dim U = 1}, L := {U ≤ F4; dim U = 2} Linear Codes over Finite Chain Rings and I :=⊆ (set inclusion). Projective and Afﬁne Thus G corresponds to the point set Hjelmslev K = F4(100), F4(010), F4(001), F4(111), F4(1αβ), F4(1βα) . Spaces Linear Code Constructions The geometric view is compatible with coding theory: Fine Structure • Replacing G by a different generator matrix G′ of the same Singer’s code corresponds to a collineation of PG 2 . Theorem ( , F4) Arcs and • Changing the order of the points and/or the coordinate Blocking Sets vectors corresponds to a code isomorphism of F6. Hyperovals 4 and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas d = 4 (or “all 3 × 3-submatrices of G are invertible”) Honold translates into the following geometric property: Motivation No three points of K are collinear. Finite Chain Rings Such point sets K are known as maximal (6, 2)-arcs or Modules over Finite Chain hyperovals. Rings Linear Codes Ovals and hyperovals over Finite Chain Rings An oval (hyperoval) in a projective plane of order q is a set Projective and Afﬁne of q + 1 points (resp., q + 2 points) meeting every line in at Hjelmslev Spaces most 2 points. Linear Code Ovals have unique tangents (1-lines) in each of their points. Constructions Hyperovals have no tangents at all (i.e. meet every line in Fine Structure

Singer’s either 0 or 2 points). Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Using geometry to compute the Thomas weight distribution of a linear Honold code Motivation n Weight distribution of C ⊆ Fq Finite Chain Rings The sequence (A0, A1,..., An) deﬁned by Modules over Ai := # c ∈ C; wHam(c)= i . Finite Chain Rings k Now suppose G = (g1|g2| . . . |gn) and x ∈ Fq \{0}. Linear Codes over Finite Observation Chain Rings ⊥ wHam(xG)= wHam (x · g1,..., x · gn) = #{i; Fqgi ∈/ x }, Projective and ⊥ n Afﬁne where L := x = {y ∈ Fq; x · y = 0} is a line of PG(2, Fq ). Hjelmslev Spaces In other words, a line L of PG(2, Fq) meeting K in n − w points

Linear Code contributes exactly q − 1 codewords of weight w. Constructions Application Fine Structure

Singer’s Hyperovals in PG(2, F4) have 15 secants (2-lines) and 6 passants Theorem (external or 0-lines). Hence the Hexacode has weight distribution Arcs and Blocking Sets i 0123 4 5 6 Hyperovals and Ovals Ai 1 0 0 0 45 0 18 Linear Codes and Geometry over Finite Chain Rings The Heptacode over Z4

Thomas Twin brother of the Hexacode Honold Linear codes over a finite ring R are defined in the same way as n for finite fields, except that “vector subspace of Fq” is replaced by Motivation n n “submodule of either RR (“left linear code”) or R (“right linear Finite Chain R Rings code”).

Modules over Finite Chain “Linear over Z4” means just “closed under addition mod 4”. Rings Observation Linear Codes 2 over Finite The Gray map γ : Z4 → F2, 0 7→ 00, 1 7→ 10, 2 7→ 11, 3 7→ 01 Chain Rings maps every linear [n, k, d] code over Z4 onto a (linear or Projective and nonlinear) binary 2n 2k d code with essentially the same Afﬁne [ , , ] Hjelmslev encoding and decoding complexity. These codes are sometimes Spaces better than the best binary linear codes. Linear Code Constructions Caution Fine Structure The observation is only true if we use the preimage Singer’s n Theorem wHam ◦ γ : Z4 → R of the Hamming weight as the corresponding weight for codes over . This is called the Lee weight. Arcs and Z4 Blocking Sets x ∈ Z4 0 1 2 3 Hyperovals and Ovals wLee(x) 0 1 2 1 Linear Codes The Heptacode is the linear code over Z4 generated by and Geometry over Finite 1001231 Chain Rings G = 0101123 Thomas   Honold 0011312 3  Motivation It is a [7, 3, 6] (or (7, 4 , 6)) code and corresponds (via the Gray

Finite Chain map) to a nonlinear binary [14, 6, 6] code. Linear binary codes Rings with these parameters do not exist. Modules over Finite Chain The columns of G deﬁne a hyperoval in the projective Hjelmslev Rings plane PHG(2, Z4) over Z4. Linear Codes over Finite Deﬁnition of PHG 2 Chain Rings ( , Z4)

Projective and The plane PHG(2, Z4) is deﬁned as the incidence structure Afﬁne with Hjelmslev (P, L, ⊆) Spaces 3 ∼ 3 ∼ Linear Code P := {U ≤ Z4; U = Z4}, L := {U ≤ Z4; U = Z4 ⊕ Z4}. Constructions

Fine Structure The points are thus the cyclic modules Z4(x1, x2, x3) with at least Singer’s one entry x1, x2, x3 equal to ±1, and the lines are generated by Theorem two linearly independent such vectors. Lines can also be Arcs and Blocking Sets described as point sets ⊥ Hyperovals Z4(x, y, z); ax + by + cz = 0 = Z4(a, b, c) for some point and Ovals Z4(a, b, c). Linear Codes and Geometry over Finite Chain Rings Properties of PHG(2, Z4)

Thomas Honold

Motivation • Tow distinct points p1, p2 ∈P can be incident with either one

Finite Chain or two lines. In the latter case the points (as well as the Rings corresponding lines L1, L2) are said to be neighbours (written Modules over as p1 ¨ p2 resp. L1 ¨ L2). Finite Chain Rings • Neighborhood corresponds to equality mod 2 and deﬁnes an Linear Codes over Finite equivalence relation on both P and L. Chain Rings

Projective and • There are 28 points and 28 lines (falling into 7 neighbour Afﬁne classes of size 4). Each line has 6 points, and each point is Hjelmslev Spaces on 6 lines.

Linear Code Constructions • The quotient structure induced on the neighbour classes [x],

Fine Structure [L] of points resp. lines is isomorphic to the Fano plane ¨ Singer’s PG(2, F2). We write x L if [x] is incident with [L]. (If Theorem L = (a, b, c)⊥, this means ax + by + cz ≡ 0 (mod 2).) Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Type, Spectrum, Weight Thomas Distribution Honold Deﬁnition n Motivation (i) The type of x ∈ Z4 is the integer triple (a0, a1, a2), where Finite Chain a0, a1, a2 count the entries of x equal to ±1, 2, resp. 0. (This Rings is sometimes also written as (±1)a0 2a1 0a2 . Modules over n Finite Chain (ii) The (symmetrized) weight enumerator of a code C ⊆ Z4 is Rings

Linear Codes a0 (c) a1(c) a2(c) over Finite AC (X0, X1, X2)= X0 X1 X2 . Chain Rings c∈C Projective and X Afﬁne Remarks Hjelmslev Spaces • Clearly AC encodes all information necessary to compute

Linear Code the Lee weight distribution of C (Hamming weight distribution Constructions of the binary image of C). Fine Structure • As in the classical case, linear codes over can be Singer’s Z4 Theorem associated with point (multi-)sets in projective Hjelmslev Arcs and spaces over Z4, and weight enumerators can be computed Blocking Sets from geometric information about the corresponding point Hyperovals and Ovals sets. Linear Codes Example (Weight enumerator of the Heptacode) and Geometry over Finite First we verify that no 3 points of K, the set of points derived from Chain Rings

Thomas Honold 1001231 G = 0101123 , Motivation 0011312 Finite Chain   Rings are collinear. (Since no two points of K are neighbours, the Modules over Finite Chain determinant test can be used for this.) Rings K K Linear Codes Since #{L; p ∈ L} = 6 = | |− 1, this implies that has only 7 over Finite 2-lines and 0-lines. There are = 21 secants and hence Chain Rings 2 28 − 21 = 7 passants. Projective and Affine Hjelmslev Definition Spaces The K-type of a line L is the integer triple (b0, b1, b2) defined by Linear Code Constructions b0 = #{p ∈ K; p 6¨ L}, Fine Structure b = #{p ∈ K; p ¨ L, p ∈/ L}, Singer’s 1 Theorem b2 = #{p ∈ K; p ∈ L} = |K ∩ L| . Arcs and Blocking Sets If L = (a, b, c)⊥, the K-type of L is equal to the type of the two Hyperovals and Ovals codewords ±(a, b, c)G. Linear Codes and Geometry over Finite Chain Rings

Thomas In our case the line types are: Honold type Motivation 2-line (±1)42102 Finite Chain 4 3 Rings 0-line (±1) 2 Modules over Finite Chain 4 2 4 3 Rings This gives a contribution of 2(21X0 X1X2 + 7X0 X1 ) to the weight

Linear Codes enumerator. over Finite Chain Rings The torsion code of the Heptacode has weight enumerator 7 4 3 Projective and X2 + 7X1 X2 (Simplex code over {0, 2}). Afﬁne Hjelmslev Hence the ﬁnal result is Spaces Linear Code 7 4 3 4 2 4 3 Constructions AC (X1, X2, X3)= X2 + 7X1 X2 + 42X0 X1X2 + 14X0 X1 . Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Recall the following facts about a ring R. Thomas • The (Jacobson) radical N rad R is deﬁned as the Honold = intersection of all maximal left ideals of R. The set N is a Motivation two-sided ideal of R (and equals the intersection of all Finite Chain maximal right ideals of R). Rings m Modules over • If R is ﬁnite (or, more generally, left Artinian) then N = 0 for Finite Chain Rings some integer m > 0.

Linear Codes over Finite • R is said to be a local ring if R has exactly one maximal left Chain Rings ideal. In this case, R/N is a division ring and R \ N is the set Projective and of units (invertible elements) of R. Affine Hjelmslev ∼ Spaces • A finite ring R is local iff R/N = Fq is a finite field (by

Linear Code Wedderburn’s Theorem). Constructions

Fine Structure • R is said to be a left chain ring if its lattice of left ideals is a chain (i.e., the left ideals of R are linearly ordered w.r.t. ). Singer’s ⊆ Theorem Right chain rings are deﬁned in an analogous manner. Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Theorem Thomas Honold For a ﬁnite ring R with radical N the following are equivalent:

Motivation Finite Chain (i) R is a left chain ring. Rings

Modules over (ii) The principal left ideals of R form a chain. Finite Chain Rings (iii) R is a local ring, and N = Rθ is left principal; Linear Codes over Finite (iv) R is a right chain ring. Chain Rings Moreover, if R satisfies these conditions, then every proper Projective and i i i Affine ideal of R has the form N = Rθ = θ R for some integer Hjelmslev Spaces i > 0. Linear Code Constructions Definition (Finite chain ring) Fine Structure A finite ring R is called a chain ring if it satisfies the Singer’s Theorem equivalent properties of the preceding theorem. Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes Proof of the theorem. and Geometry over Finite If N = {0}, then R =∼ Fq is a finite field and satisfies (i)–(iv). Chain Rings From now on we assume N 6= {0} and hence N2 ( N. Thomas Honold (i)⇒(iii): Clearly R is local. 2 Motivation Consider the quotient module V := N/N .

Finite Chain NV = VN = 0, so V is both a left and a right vector space over Rings R/N =∼ Fq (necessarily of the same dimension). Modules over 2 Finite Chain The left ideals between N and N correspond to subspaces of V . Rings Since they form a chain, we must have dim V = 1. Linear Codes 2 over Finite Let θ ∈ N \ N . Chain Rings Rθ ⊆ N Projective and 2 =⇒ Rθ = N Afﬁne Rθ * N Hjelmslev Spaces (iii)⇒(i),(iv): The crucial step is to show N = θR. Linear Code N = Rθ =⇒ dim V = 1 R/N =⇒ N = θR + N2 Constructions θ ∈ N \ N2 Fine Structure

Singer’s This gives Theorem 2 2 Arcs and N = θR + N = θR + (θR + N )N Blocking Sets 3 m Hyperovals = θR + N = · · · = θR + N = θR. and Ovals Linear Codes and Geometry over Finite Chain Rings Proof cont’d. Thomas i i i Honold It follows that N = Rθ = θ R for i = 1, 2,... .

Motivation We have the chain of two-sided ideals

Finite Chain 0 2 m−1 m Rings R = N ) N ) N ) · · · ) N ) N = {0}. (⋆) Modules over Finite Chain Rings Suppose now 0 6= a ∈ R. There exists i ∈{0, 1,..., m − 1} such that a Ni Ni+1 R i R i+1, i.e. Linear Codes ∈ \ = θ \ θ over Finite Chain Rings a = uθi where u ∈ R \ N is a unit. Projective and Afﬁne i Hjelmslev Hence Ra = Rθ , and thus every left ideal of R belongs to the Spaces chain (⋆). Linear Code Constructions We have proved that property (iii) is left-right symmetric and Fine Structure implies (i). Hence (iii)⇒(iv) is true as well. Singer’s Theorem The remaining implications are trivial. Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Properties of Finite Chain Rings

Thomas Honold

Motivation With every ﬁnite chain ring R we associate the following pair q m of integers: Finite Chain ( , ) Rings q The order of the residue class ﬁeld R/N (i.e. Modules over Finite Chain R/N =∼ Fq) Rings

Linear Codes m The index of nilpotency of N (equal to the over Finite composition length of the regular module R) Chain Rings R Projective and We have the following counting formulas: Afﬁne Hjelmslev m m−1 × m m−1 m−1 Spaces • |R| = q , |N| = q , |R | = q − q = (q − 1)q Linear Code i m−i i i Constructions • N = q , R/N = q for 0 ≤ i ≤ m Fine Structure For the proof it sufﬁces to observe that

Singer’s i i+1 i i+1 ∼ Theorem N /N = Rθ /Rθ = R/N as a vector space over R/N.

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Galois Rings r Let q = p > 1 be a prime power and h(X) ∈ Zpm [X] a monic Thomas Honold polynomial of degree r which is irreducible modulo p (a so-called basic irreducible polynomial or Galois polynomial). Motivation

Finite Chain Definition Rings m m The ring GR(q , p ) := Zpm [X]/ h(X) is called Galois ring of Modules over order qm and characteristic pm. Finite Chain Rings Properties of R = GR(qm, pm) Linear Codes over Finite • R is a finite commutative chain ring with radical N = Rp, Chain Rings invariants (q, m) and characteristic pm. Projective and Affine • The isomorphism type of R does not depend on the Hjelmslev Spaces particular choice of the polynomial h(X). m Linear Code • A finite chain ring with invariants (q, m) and characteristic p Constructions is isomorphic to R. Fine Structure • Every automorphism of R/N can be lifted in a unique way to Singer’s an automorphism of R. Thus Aut R =∼ Aut Fq =∼ (Zr , +). Theorem Remark Arcs and m rm m Blocking Sets For the Galois ring of order q = p and characteristic p two different m m Hyperovals notations are widely used: GR(q , p ) (due to Raghavendran [?]) and and Ovals GR(pm, r) (due to Janusz [?]). Linear Codes and Geometry over Finite Chain Rings Representation Theorem Thomas Definition Honold Suppose S is a Galois ring. Motivation (i) For σ ∈ Aut S the skew polynomial ring S[X; σ] is defined in Finite Chain Rings the same way as the polynomial ring S[X], except that Modules over Xa = σ(a)X for a ∈ S. Finite Chain Rings (ii) A polynomial of the form Linear Codes g(X)= X k + p(g X k−1 + · · · + g X + g ) ∈ S[X; σ] where over Finite k−1 1 0 Chain Rings g0 ∈ S \ pS is said to be an Eisenstein polynomial. Projective and Affine Hjelmslev Theorem Spaces Suppose R is a finite chain ring with invariants (q, m) and Linear Code characteristic ps (1 s m). Let S GR qs ps . Then there Constructions ≤ ≤ = ( , ) exist (unique) integers k, t satisfying m = (s − 1)k + t, 1 ≤ t ≤ k Fine Structure (k = t = m ifs = 1), an automorphism σ ∈ Aut S and a (possibly Singer’s Theorem nonunique) Eisenstein polynomial g(X) ∈ S[X; σ] such that Arcs and Blocking Sets R =∼ S[X; σ]/ g(X), ps−1X t . Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Case m = 2

Thomas Corollary Honold Suppose R is a finite chain ring with residue class field Fq, r 2 Motivation q = p , and nilpotency index m = 2. Then |R| = q , and either Finite Chain 2 2 2 Rings (i) R has characteristic p and R =∼ GR(q , p ), or Modules over Finite Chain (ii) R has characteristic p and there exists σ ∈ Aut Fq such that Rings 2 R =∼ Fq[X; σ]/(X ). Linear Codes over Finite Thus there are exactly r + 1 isomorphism types of such rings. Chain Rings 2 2 Two of them, Gq := GR(q , p ) and Projective and 2 2 Affine Sq := Fq[X; id]/(X )= Fq[X]/(X ), are commutative. Hjelmslev Spaces Example Linear Code Constructions The chain rings of order ≤ 16 with m = 2 are

Fine Structure 2 |R| = 4 G2 = Z4, S2 = Z2[X]/(X ) Singer’s 2 Theorem |R| = 9 G3 = Z9, S3 = Z3[X]/(X ) 2 2 Arcs and |R| = 16 G4 = Z4[X]/(X + X + 1), S4 = F4[X]/(X ), and Blocking Sets 2 2 T4 = F4[X; a 7→ a ]/(X ) with multiplication Hyperovals 2 and Ovals (a0 + a1X)(b0 + b1X)= a0b0 + (a0b1 + a1b0)X. Linear Codes and Geometry over Finite Chain Rings Consider first the special case R = Zpm .A Zpm -module is Thomas m Honold the same thing as an Abelian group of exponent dividing p (i.e. the additively written abelian group G satisfies Motivation pmG = {pmx; x ∈ G} = {0}). Finite Chain Rings The following is well-known: Modules over Finite Chain Rings Theorem Linear Codes For every finite abelian p-group G there exists a unique over Finite Chain Rings sequence of integers λ1 ≥ λ2 ≥···≥ λr ≥ 1 such that Projective and Affine ∼ λ λ λ Hjelmslev G = Zp 1 ⊕ Zp 2 ⊕···⊕ Zp r . Spaces

Linear Code m Constructions The group G satisﬁes p G = {0} iff λ1 ≤ m. Fine Structure Now we shall generalize this theorem to modules over Singer’s Theorem arbitrary ﬁnite chain rings (and thereby reprove it). Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Some Notation R denotes a ﬁnite chain ring with invariants (q, m) and radical Thomas 2 Honold N = Rθ (where θ ∈ N \ N is chosen arbitrarily). All modules considered will be ﬁnite. Motivation

Finite Chain Definition Rings For a left R-module RM and x ∈ M we say Modules over Finite Chain (i) x has period θi , if i ∈{0, 1,..., m} is the smallest integer Rings satisfying θi x = 0; Linear Codes over Finite Chain Rings (ii) x has height i, if i ∈{0, 1,..., m} is the largest integer i Projective and satisfying x = θ y for some y ∈ M. Affine Hjelmslev We also define M× := x ∈ M; period(x)= θm (“units of M”). Spaces R Linear Code Constructions Remarks Fine Structure • 0 ∈ M is the only element with period θ0 = 1 (resp., height m) Singer’s Theorem • If x has period θi , then R → Rx, r 7→ rx has kernel Rθi = Ni . Arcs and ∼ i ∼ Blocking Sets Hence Rx = R/N as left R-modules, and Rx = R iff x M×. Hyperovals ∈ and Ovals Linear Codes Further we define and Geometry over Finite Chain Rings θi M := {θi x; x ∈ M}, Thomas i i Honold M[θ ] := {x ∈ M; θ x = 0}, i−1 Motivation Vi := M[θ] ∩ θ M.

Finite Chain i i Rings All these sets are submodules of RM (since Rθ = θ R). Modules over Finite Chain We have the submodule chains Rings Linear Codes M = θ0M ⊇ θ1M ⊇ θ2M ⊇···⊇ θmM = {0}, over Finite Chain Rings (upper Loewy series of RM) Projective and Afﬁne M = M[θm] ⊇ M[θm−1] ⊇···⊇ M[θ0]= {0}, Hjelmslev Spaces (lower Loewy series of RM) Linear Code Constructions whose successive quotients are vector spaces over R/N =∼ Fq Fine Structure (since they are annihilated by θ), Singer’s Theorem and the chain of R/N-spaces Arcs and Blocking Sets M[θ]= V1 ⊇ V2 ⊇···⊇ Vm ⊇{0} Hyperovals (Ulm-Kaplansky series of M) and Ovals R Linear Codes and Geometry Proposition over Finite Chain Rings i−1 i (i) For 1 ≤ i ≤ m we have dimR/N θ M/θ M = Thomas dim M[θi ]/M[θi−1] = dim M[θ] ∩ θi−1M . Honold R/N R/N i−1 Motivation (ii) The integers µi := dimR/N M[θ] ∩θ M , 1 ≤ i≤ m Finite Chain (Ulm-Kaplansky invariants of RM) satisfy µ1 ≥ µ2 ≥···≥ µm Rings and µ1 + µ2 + · · · + µm = logq |M|. Modules over Finite Chain Rings

Linear Codes Proof. over Finite (i) The map θi−1M → θi M, x 7→ θx induces an (additive) Chain Rings isomorphism θi−1M/ M[θ] ∩ θi−1M =∼ θi M. Hence Projective and Afﬁne Hjelmslev θi−1M Spaces = θi M and θi−1M/θi M = M[θ] ∩ θi−1M . M i−1M Linear Code | [θ ] ∩ θ | Constructions i i−1 Fine Structure Similarly, M[θ ] → M[θ], x 7→ θ x induces an isomorphism i i−1 i−1 Singer’s M[θ ]/M[θ ] → M[θ] ∩ θ M, proving the second equality. Theorem

Arcs and (ii) The inequality µi ≥ µi+1 follows from µi = dimR/N (Vi ) and Blocking Sets Vi ⊇ Vi+1. Part (i) and the upper Loewy series give m Hyperovals |M| = θi−1M/θi M = qµ1+···+µm . and Ovals i=1 Q

Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Example n Motivation Let R = Z4 = {0, 1, 2, 3}, M = Z4, θ = 2 (the only choice for θ). Finite Chain The possible periods (heights) of x ∈ n are 1, 2, 4 (resp. 0, 1, 2). Rings Z4 n Modules over x ∈ Z4 has period 1 (height 2) iff x = 0 Finite Chain x has period 2 (height 1) iff 2x = 0 ∧ x 6= 0 iff x ∈{0, 2} and at Rings i least one xi = 2. Linear Codes over Finite x has period 4 (height 0) iff at least one xi ∈{1, 3}. Chain Rings We have n[2]= 2 n, so both Loewy series coincide (a property Projective and Z4 Z4 Afﬁne of free modules in general). Hjelmslev Spaces The factors are n ∼ n Linear Code 2Z4 = Z2 (divide the entries of x by 2) Constructions n n ∼ n Z4/2Z4 = Z2 (read the entries of x modulo 2) Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Example Honold Let C be the linear code of even length n over Z4 generated by Motivation 1 1 1 . . . 1 Finite Chain Rings 2 2 0 . . . 0   Modules over .. . Finite Chain 0 2 2 . . Rings    ......  Linear Codes  . . . . 0   over Finite 0 . . . 0 2 2 Chain Rings   Projective and   Afﬁne 2C = {00 . . . 0, 22 . . . 2} Hjelmslev n n Spaces C[2] is either 2Z4 (if n is odd) or the even-weight subcode of 2Z4 Linear Code (if n is even) Constructions So the Loewy series C ⊃ 2C ⊃{0} and C ⊃ C[2] ⊃{0} are Fine Structure different, but its factors C/2C =∼ C[2] and C/C[2] =∼ 2C are the Singer’s Theorem same.

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Partitions of integers Chain Rings Deﬁnition Thomas Honold An (integer) partition is a sequence λ = (λ1, λ2,... ) with λi ∈ Z, λ1 ≥ λ2 ≥··· , and λi = 0 for all but ﬁnitely many i. The numbers Motivation λi > 0 are called the parts of λ, and |λ| = λ1 + λ2 + · · · is called Finite Chain Rings the weight of λ. If |λ| = n, we say that λ is an (unordered)

Modules over partition of n and write λ ⊢ n. Finite Chain Rings Trailing zeros are usually suppressed, e.g. (2, 1, 1, 0, 0,... ) is

Linear Codes written as (2, 1, 1), or as 4 = 2 + 1 + 1. over Finite Chain Rings A partition λ is often visualized by an (empty) Young tableaux Tλ, Projective and shown here for λ = (6, 6, 4, 3, 2, 2, 1, 1, 1) ⊢ 26: Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem Arcs and Think of Tλ as the union of all unit squares in the Euclidean plane Blocking Sets whose upper right corners have coordinates (i, j) where i ≥ 1 and Hyperovals and Ovals 1 ≤ j ≤ λi . Linear Codes Deﬁnition and Geometry ′ over Finite Let λ be a partition. The conjugate λ of λ is the partition whose Chain Rings Young tableaux Tλ′ is obtained from Tλ by a reﬂection at the line Thomas Honold y = x.

Motivation Properties Finite Chain • If n then also ′ n. Rings λ ⊢ λ ⊢ ′ ′ Modules over • The parts of λ are λ = |{j; λj ≥ i}|. Finite Chain i Rings ′ ′ • The largest part λ1 of λ is equal to the number of parts of λ. Linear Codes over Finite Chain Rings Example Projective and ′ Afﬁne The conjugate of λ = (6, 6, 4, 3, 2, 2, 1, 1, 1) is λ = (9, 6, 4, 3, 2, 2). Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Module Classiﬁcation Theorem

Thomas Honold

Motivation Theorem Finite Chain For any finite module RM there exists a uniquely determined Rings partition λ ⊢ log |M| into parts λi ≤ m such that Modules over q Finite Chain Rings λ1 λ2 λr RM =∼ R/N ⊕ R/N ⊕···⊕ R/N . Linear Codes over Finite Chain Rings (Here r denotes the number of parts of λ.) Projective and Affine Remarks Hjelmslev Spaces • The theorem holds mutatis mutandis for right modules M . Linear Code R Constructions • The theorem says in particular that every finite R-module is a Fine Structure direct sum of cyclic R-modules. Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Proof of the theorem. Motivation r ∼ λi Uniqueness part: Suppose RM = i=1 R/N . Finite Chain λ Rings The upper Loewy series of R/N i is clearly L Modules over Finite Chain R/Nλi ) N/Nλi ) N2/Nλi ) · · · ) Nλi /Nλi = {0} Rings

Linear Codes over Finite with all successive quotients 1-dimensional over R/N. Chain Rings Hence j−1 j ′ Projective and dimR N θ M/θ M = |{i; λi ≥ j}| = λ Afﬁne / j Hjelmslev ′ Spaces This shows that λ (and hence λ) is uniquely determined by R M. Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals i Linear Codes Existence part: Since any cyclic R-module is isomorphic to R/N and Geometry over Finite for some i ≤ m, it sufﬁces to prove that RM is a direct sum of Chain Rings cyclic R-modules. We use induction on |M|. Thomas Honold Assume M 6= {0} (otherwise the assertion is clear) and choose λ x1 ∈ M of maximal period θ 1 . By induction, Motivation

Finite Chain Rings M/Rx1 = Ry2 ⊕···⊕ Ryr . (⋆)

Modules over Finite Chain Let xi ∈ M such that yi = xi + Rx1 (2 ≤ i ≤ r). Then Rings

Linear Codes M = Rx + Rx + · · · + Rxr . (⋆⋆) over Finite 1 2 Chain Rings

Projective and To make (⋆⋆) a direct sum decomposition, the xi ’s must be chosen Afﬁne in a special way. Hjelmslev Spaces λi λi Suppose yi has period θ , i.e. θ xi ∈ Rx1 and λi is the smallest Linear Code Constructions such integer.

Fine Structure λ λ There exists r ∈ R such that θ i xi = θ i ax1. (Otherwise the period Singer’s of x would be smaller than the period of x .) Theorem 1 i Then λi x ax 0, and so x ax has period λi . Arcs and θ ( i − 1)= i − 1 θ Blocking Sets Replacing xi by xi − ax1, we may assume that the xi ’s in (⋆⋆) have Hyperovals λi and Ovals the same period θ as the yi ’s in (⋆). Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation Proof cont’d. Finite Chain Rings Now suppose

Modules over Finite Chain a1x1 + a2x2 + · · · + ar xr = 0 where ai ∈ R Rings Linear Codes Modulo Rx this reads a y a y 0, and ( ) gives over Finite 1 2 2 + · · · + r r = ⋆ Chain Rings a2y2 = · · · = ar yr = 0. Projective and Afﬁne Since xi has the same period as yi , we conclude Hjelmslev Spaces a2x2 = · · · = ar xr = 0, and further a1x1 = 0.

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes Definition and Geometry r over Finite ∼ λi The partition λ ⊢ logq |M| such that RM = i=1 R/N is called Chain Rings ′ the shape (or type) of RM. The conjugate partition λ , which Thomas ′ i−1 i L Honold satisfies λi = dimR/N θ M/θ M , is called the conjugate shape ′ of RM. The integer r = λ1 (number of nonzero summands in a Motivation direct sum decomposition of RM into cyclic R-modules) is called Finite Chain Rings the rank of RM (and denoted by rk M). Modules over Definition Finite Chain Rings A sequence x1, x2,..., xs of elements of RM is said to Linear Codes over Finite (i) independent if a1x1 + a2x2 + · · · + asxs = 0 with ai ∈ R Chain Rings implies ai xi = 0 for 1 ≤ i ≤ s; Projective and Affine (ii) a basis of RM if x1,..., xr are independent and generate RM; Hjelmslev Spaces (iii) linearly independent if a1x1 + a2x2 + · · · + asxs = 0 with Linear Code Constructions ai ∈ R implies ai = 0 for 1 ≤ i ≤ s. Fine Structure Singer’s Remarks Theorem

Arcs and • Properties (ii) is equivalent to RM = Rx1 ⊕···⊕ Rxs. Blocking Sets × • Property (iii) is equivalent to (i) and xi ∈ M for 1 ≤ i ≤ s. Hyperovals and Ovals Linear Codes and Geometry over Finite Example Chain Rings Consider the linear codes C , C over generated by Thomas 3 4 Z4 Honold 1 1 1 1 1 1 1 Motivation 2 2 0 0 G = 2 2 0 resp. G = . Finite Chain 3 4 0 2 2 0 Rings 0 2 2 Modules over 0 0 2 2 Finite Chain     Rings   The rows of G3 are independent, hence a basis of C3. Linear Codes over Finite The ﬁrst 3 rows of G4 form a basis of C4 (since they are Chain Rings independent, and the last row is a linear combination of the 1st Projective and Afﬁne and 2nd row). Hjelmslev Spaces Hence both modules have the same shape λ = (2, 1, 1) with Linear Code Young diagram Constructions

Fine Structure

Singer’s Theorem

Arcs and 2+1+1 Blocking Sets rank equal to 3 and cardinality |C3| = |C4| = 2 = 16.

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Example

Motivation The linear code over Z4 generated by Finite Chain Rings 1001231 Modules over Finite Chain 0101123 Rings 0011312 Linear Codes over Finite   Chain Rings has shape λ = (2, 2, 2) with Young diagram Projective and Afﬁne Hjelmslev Spaces

Linear Code Constructions 2+2+2 Fine Structure rank 3 and cardinality 2 = 64.

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Free Modules ∼ k Thomas Recall that RM is free if RM = R for some integer k ≥ 0. (The Honold integer k is the rank of M.) Equivalently, RM has “rectangular”

Motivation shape (m, m,..., m). Finite Chain Free modules are important, since they are the ambient spaces Rings for both linear codes and projective Hjelmslev geometries over R. Modules over Finite Chain Rings Deﬁnition

Linear Codes Let RM be free of rank k. The projective geometry PG(R M) is the over Finite lattice of all submodules of M. A free submodule U M of Chain Rings R ≤ R rank 1 (rank k − 1, rank s + 1) is said to be a point (resp. Projective and Afﬁne hyperplane, s-ﬂat) of PG(RM). Hjelmslev Spaces Important problems Linear Code Constructions

Fine Structure • Determine the number of points (hyperplanes, s-ﬂats, or in

Singer’s general λ-shaped submodules) of PG(RM). Theorem

Arcs and • How many hyperplanes contain a given point (s-ﬂat, or Blocking Sets λ-shaped submodule)? Hyperovals and Ovals Linear Codes and Geometry over Finite Theorem Chain Rings Suppose R M is free of rank k, and U is a (not necessarily free) Thomas submodule of RM of shape λ and rank s. Honold 1 For every basis x1,..., xs of U there exists a basis y1,..., yk Motivation of M such that xi ∈ Ryi for 1 ≤ i ≤ s. (Bases of U and M Finite Chain Rings related in this way are called stacked bases.) Modules over 2 The quotient module M U has shape Finite Chain / Rings (m − λk , m − λk−1,..., m − λ1). Linear Codes ∗ over Finite 3 If U 6= ∅ then U is the sum of its free rank 1 submodules Chain Rings (i.e., points). Projective and Afﬁne 4 ∗ Hjelmslev If (M/U) 6= ∅ then U is the intersection of all free rank k − 1 Spaces submodules of R M (i.e., hyperplanes) containing U. Linear Code Constructions Fine Structure Remark Singer’s M/U Theorem Part (2) says that the shape of M/U is

Arcs and the complement of λ in the Blocking Sets k × m-rectangle corresponding to M. U Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Lemma i m−i If RM is free then M[θ ]= θ M for 0 ≤ i ≤ m. Thomas Honold Proof. Motivation Suppose M is free of rank k. For the shape λ of RM this means Finite Chain Rings ′ ′ λ1 = · · · = λk = m, λ1 = · · · = λm = k. Modules over Finite Chain i i 1 ′ k Rings So M[θ ]/M[θ − ] = qλi = q for 1 ≤ i ≤ m, and hence Linear Codes M[θi ] = qki for 0 ≤ i ≤ m. over Finite Chain Rings By a similar argument using the upper Loewy series of RM we Projective and have θi M = qk(m−i). Afﬁne Hjelmslev i m−i Spaces Hence M[ θ ] = θ M for 0 ≤ i ≤ m. Together with θm−i M ⊆ M [θi ] this gives M[θi ]= θm−i M for 0 ≤ i ≤ m. Linear Code Constructions Fine Structure Remark Singer’s Suppose R is a proper chain ring (i.e. m ≥ 2). If there exists Theorem i m−i i ∈{1, 2,..., m − 1} such that M[θ ]= θ M, then RM is Arcs and Blocking Sets necessarily free.

Hyperovals and Ovals Linear Codes Proof of the theorem. and Geometry over Finite (1) Suppose M is free of rank k, U ≤ M has shape λ, and Chain Rings λ x1,..., xs is a basis of U with xi of period θ i . Thomas λi m−λi m−λi Honold M[θ ]= θ M ⇒ ∃yi ∈ M with xi = θ yi (1 ≤ i ≤ s). m y1,..., ys are linearly independent (since their period is θ ), Motivation hence may be extended to a basis y1,..., yk of M (e.g., by ﬁrst Finite Chain extending m−1y m−1y to a basis of the R N-space M Rings θ 1,...,θ s / [θ] and then proceeding as before). Modules over Finite Chain Rings (2) With yi as in (1) we have

Linear Codes k−s over Finite M/U =∼ Ry1/Rx1 ⊕···⊕ Rys/Rxs ⊕ R Chain Rings ∼ R Nm−λ1 R Nm−λs R Nm R Nm Projective and = / ⊕···⊕ / ⊕ / ⊕···⊕ / Afﬁne m−λ1 m−λk Hjelmslev = R/N ⊕···⊕ R/N . Spaces × × Linear Code (3) U 6= ∅ is equivalent to λ1 = m, i.e. x1 ∈ M . Constructions × × × For j ≥ 2, if xj ∈/ U then x1 + xj ∈ U , so U is generated by U . Fine Structure ∗ Singer’s (4) (M/U) 6= ∅ is equivalent to λk = 0. In this case U is Theorem contained in the hyperplane H = Ry1 + · · · + Ryk−1. Arcs and For z H U, z r y r y with m−λj r , say, let H′ Blocking Sets ∈ \ = 1 1 + · · · + k−1 k−1 θ ∤ j be the hyperplane obtained from H by replacing y by y + θλj y . Hyperovals j j k and Ovals One easily checks that U ⊆ H′, but z ∈/ H′. Linear Codes and Geometry over Finite Chain Rings Embedding into Free Modules

Thomas Honold Deﬁnition ′ ′ Motivation Suppose R M, RM are modules over R. A mapping φ: RM → RM Finite Chain is said to be semilinear with associated ring homomorphism Rings σ : R → R if φ(x + y)= φ(x)+ φ(y) and φ(rx)= σ(r)φ(x) for all Modules over Finite Chain x, y ∈ M and r ∈ R. Rings Linear Codes Remarks over Finite Chain Rings • Semilinear mappings are important in geometry (−→ Projective and Afﬁne Fundamental Theorem of Projective Hjelmslev Geometry) Hjelmslev Spaces • If M× 6= ∅ and φ is in an embedding, then σ ∈ Aut R and σ is Linear Code uniquely determined by . In this case preserves free Constructions φ φ modules. Fine Structure Singer’s • The group of all R-linear (R-semilinear) isomorphisms Theorem φ: RM → RM is denoted by GL(RM) (resp. ΓL(RM)). Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry Proposition over Finite Chain Rings For every module RM there exists a minimal free module RH

Thomas containing RM. More precisely, there exists an R-linear Honold embedding ι: RM → RH such that no proper free submodule of H contains M . Motivation R ι( )

Finite Chain Rings Proof.

Modules over The minimality of RH is equivalent to rk H = rk M. In this case RH Finite Chain contains a submodule of the same shape as RM. Rings

Linear Codes over Finite Theorem × Chain Rings Let RM be a ﬁnite module with M 6= ∅ and RH a minimal free Projective and module containg RM. Afﬁne Hjelmslev Spaces (i) Any semilinear embedding of RM into a free module RF can

Linear Code be extended to a semilinear embedding of RH into RF. Constructions (ii) If M M′ is a semilinear isomorphism and H′ a Fine Structure φ: R → R R minimal free module containing M′, then there exists a Singer’s R ′ Theorem semilinear isomorphism φ: RH → RH which extends φ. Arcs and Blocking Sets The theorem allows us to restricte attention to free R-modules in Hyperovals and Ovals most situations. Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Main idea of the proof. Rings Reduction to the R-linear case: Modules over σ Finite Chain φ: RM → RF is semilinear iff φ is linear as a map from RM to RF , Rings σ where RF has scalar multiplication deﬁned by rx := σ(r)x. Linear Codes over Finite Chain Rings The proof can be completed using either the existence of stacked

Projective and bbases for RM and RH or the fact that RH is an injective hull of Afﬁne RM. Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Counting Formulas for Submodules Thomas First recall the following Honold

Motivation Fact An n-dimensional vector space over has exactly Finite Chain Fq Rings n n n k−1 Modules over n (q − 1)(q − q) · · · (q − q ) := Finite Chain k (qk − 1)(qk − q) · · · (qk − qk−1) Rings q Linear Codes (qn − 1)(qn−1 − 1) · · · (qn−k+1 − 1) over Finite = Chain Rings (qk − 1)(qk−1 − 1) · · · (q − 1) Projective and Affine Hjelmslev k-dimensional subspaces. Spaces In particular, an n − 1-dimensional projective geometry over Fq n Linear Code q −1 Constructions has q−1 points and the same number hyperplanes. Fine Structure Question Singer’s Theorem Let R be a finite chain ring with invariants (q, m). How many n Arcs and points (hyperplanes) does the projective geometry PG(R R ) Blocking Sets have? Hyperovals and Ovals Linear Codes and Geometry Answer n over Finite A free cyclic submodule of R has the form Rx = R(x ,..., xn) Chain Rings R 1 × where at least one xi ∈ R . Thomas Honold Normalize x by setting the first unit coordinate equal to 1. For fixed i 1 n there are Motivation ∈{ ,..., }

Finite Chain i−1 n−i (m−1)(i−1)+m(n−i) mn−m−i+1 Rings |N| |R| = q = q Modules over Finite Chain choices for x, such that x 1 and x R× for j i. Rings i = j ∈/ < Hence the number of points of PG( Rn) is Linear Codes R over Finite Chain Rings n n n mn−m−i+1 (m−1)(n−1) n−i (m−1)(n−1) q − 1 Projective and q = q q = q · . Afﬁne q − 1 Hjelmslev i=1 i=1 Spaces X X

Linear Code To count the number of hyperplanes we could Constructions

Fine Structure • either use some duality theory to conclude (hopefully) that the number of points and hyperplanes is the same, Singer’s Theorem • or prove a general counting formula for the number of Arcs and Blocking Sets µ-shaped submodules of a λ-shaped module and specialize Hyperovals this to the case at hand. and Ovals Linear Codes and Geometry over Finite Chain Rings The General Counting Formula i−1 i−1 For U ≤ RM we have U[θ] ∩ θ U ⊆ M[θ] ∩ θ M. Thomas Honold ⇒ The shapes λ, µ of RM resp. R U satisfy µ ≤ λ (part-wise).

Motivation Theorem

Finite Chain Let RM be a module of shape λ. For every partition µ satisfying Rings µ ≤ λ the module RM has exactly Modules over Finite Chain m ′ ′ Rings µ′ (λ′−µ′) λi − µi+1 αλ(µ; q) := q i+1 i i · Linear Codes ′ ′ µi − µi 1 over Finite i=1 + q Chain Rings Y Projective and submodules of shape µ. In particular, the number of free rank 1 Affine Hjelmslev submodules of RM equals Spaces ′ λ′ −1+λ′ −1+···+λ′ −1 λm Linear Code q 1 2 m−1 · . Constructions 1 Remark q Fine Structure One can memorize the 2nd formula as Singer’s Theorem follows: The top row and first column of ′ Arcs and λ (consisting of λm + m − 1 squares) Blocking Sets λ′ account for q m −1 points. Each further Hyperovals q−1 and Ovals square accounts for a factor q. Linear Codes and Geometry Proof. over Finite i−1 Chain Rings Let Vi := M[θ] ∩ θ M, and for U ≤ RM similarly U U i−1U (1 i m). Thomas i := [θ] ∩ θ ≤ ≤ Honold Then Ui ⊆ Vi , dimR/N (Vi )= λi , and dimR/N (Ui )= µi .(⋆) m λ′−µ′ Motivation There are i i+1 possible ways to choose the j=1 µ′−µ′ i i+1 q Finite Chain Rings Ulm-KaplanskyQ chain U1 ⊇ U2 ⊇···⊇ Um for RU subject to (⋆). Modules over Let x1,..., xµ′ be a basis of U1 such that for every i the Finite Chain 1 Rings subsequence x ,..., x ′ is a basis of U . 1 µi i Linear Codes µ −1 A basis y ,..., y ′ for U such that θ j y = x for all j can be over Finite 1 µ1 R j j Chain Rings ′ µ1 µj −1 ′ chosen in j 1 M[θ ] ways. Two such bases (yj ), (yj ) yield Projective and = ′ µj −1 Affine the same submodule U iff yj − yj ∈ U[θ ] for all j. Hjelmslev Q Spaces i ′ ′ i ′ ′ Now M[θ ] = qλ1+···+λi , U[θ ] = qµ1+···+µi , so there are exactly Linear Code Constructions µ′ µ′ 1 µj −1 1 µ −1 M[ θ ] P j (λ′−µ′) Fine Structure = q i=1 i i U[θµj −1] Singer’s j=1 j=1 Theorem Y Y P {j;µ ≥i+1} (λ′−µ′) P µ′ (λ′−µ′) Arcs and = q i≥1| j | i i = q i≥1 i+1 i i Blocking Sets

Hyperovals and Ovals submodules U with Ulm-Kaplansky chain U1 ⊇···⊇ Um. Linear Codes and Geometry Corollary over Finite Chain Rings Let RM be a free module of rank n. The number of submodules of M with complementary shapes resp. Thomas R µ = (µ1, µ2,...,µn) Honold µ = (m − µn, m − µn−1 ..., m − µ1) is the same. In particular, PG( M) has the same number of points and hyperplanes. Motivation R

Finite Chain Rings Proof. µ′ = (n − µ′ , n − µ′ ,..., n − µ′ ). Modules over m m−1 1 Finite Chain Rings The number of chains U1 ⊇ U2 ⊇···⊇ Um of subspaces of a ′ Linear Codes n-dimensional vector space satisfying dim Ui = n − µm+1−i equals over Finite the number of chains U ⊇ U ⊇···⊇ U of subspaces satisfying Chain Rings 1 2 m dim U = µ′ (by duality). Projective and i i m n ′ Afﬁne Hence the second factor −µi+1 is invariant under Hjelmslev i=1 µ′−µ′ i i+1 q Spaces complementation. Linear Code Q Constructions Further Fine Structure m−1 m−1 m−1 Singer’s ′ ′ ′ ′ ′ ′ Theorem µi+1(n − µi )= (n − µm−i )µm−i+1 = µj+1(n − µj ), Arcs and i=1 i=1 j=1 Blocking Sets X X X m−1 Hyperovals P ... so the ﬁrst factor q i=1 is invariant as well. and Ovals Linear Codes and Geometry over Finite Chain Rings Applications

Thomas Example Honold n ′ The module RR has conjugate shape λ = (n, n,..., n). Using Motivation the general formula, the number of points (hyperplanes) of n Finite Chain PHG(RR ) is equal to Rings

Modules over ′ ′ 1 ′ 1 ′ 1 λm n 1 n 1 n Finite Chain qλ1− +λ2− +···+λm−1− · = q − +···+ − Rings 1 1 q q Linear Codes qn − 1 over Finite = q(m−1)(n−1) · . Chain Rings q − 1 Projective and Afﬁne Example Hjelmslev n Spaces Suppose Rx is a point of PHG(R R ). A point Ry is an i-th i n Linear Code neighbour of Rx iff Ry ⊆ Rx + θ R . Constructions The module Rx + θi Rn has shape (m, m − i,..., m − i) and Fine Structure conjugate shape (n,..., n, 1,..., 1). Singer’s Theorem m−i i Arcs and Hence Blocking Sets | {z(m}−i)(| n−{z1) } 1 (m−i)(n−1) |[Rx]i | = q · = q . Hyperovals 1 q and Ovals Linear Codes Example and Geometry over Finite The number of s 1-dimensional Hjelmslev subspaces of Chain Rings − n n PHG(RR ) (free rank s submodules of RR ) is obtained by Thomas ′ ′ Honold plugging λ = (n, n,..., n) and µ = (s, s,..., s) into the general formula: Motivation

Finite Chain m ′ ′ ′ ′ ′ λ − µ s n s s n s n Rings qµi+1(λi −µi ) · i i+1 = q ( − )+···+ ( − ) · µ′ − µ′ s Modules over i=1 i i+1q q Finite Chain Y Rings n = qs(n−s)(m−1) · . Linear Codes s over Finite q Chain Rings n Projective and Now suppose U is a free rank t submodule of RR . Afﬁne Hjelmslev A submodule V ⊇ U is free of rank s iff V /U (a submodule of Spaces n n−t RR /RU =∼ RR is free of rank s − t. Linear Code Constructions ⇒ The number of s − 1-dimensional Hjelmslev subspaces of n n Fine Structure PHG(RR ) of PHG(R R ) through a ﬁxed t − 1-dimensional Singer’s Hjelmslev subspace is equal to Theorem

Arcs and (s−t)(n−t−(s−t))(m−1) n − t (s−t)(n−s)(m−1) n − t Blocking Sets q · = q · . s − t s − t Hyperovals q q and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation Remark Finite Chain The preceding formula is a special case of the following Rings Modules over Proposition Finite Chain Rings Let λ, µ partitions with µ ≤ λ ≤ (m, m,..., m). Then every Linear Codes over Finite n n Chain Rings submodule U ≤ RR of shape µ is contained in exactly α (λ; q) | {z } µ Projective and submodules V ≤ M of shape λ, where λ, µ denote the Afﬁne n Hjelmslev complementary shapes of λ, µ (i.e. the shapes of R /U resp. Spaces Rn/V). Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Example Rings 3 The Z4-module Z4 has the following number of submodules of Modules over Finite Chain each shape: Rings

Linear Codes over Finite shape Chain Rings

Projective and #submodules 7 7 1 28 42 7 28 7 Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Structure of Linear Codes

Thomas Honold n Motivation For linear codes over R (i.e. submodules C ≤ RR ) the structure theorem for R-modules says: Finite Chain Rings There exists a uniquely determined partition λ into at most n Modules over parts, all of which are ≤ m, such that RC has a basis c1,..., ck Finite Chain ′ λ1 λk Rings (k = λ1 = rk C) with corresponding periods θ ,...,θ . Linear Codes n n over Finite Since RR is free, ci has height m − λi in RR , i.e. Chain Rings m−λi m−λi m−λi Projective and ci = (θ ci1,θ ci2,...,θ cin) Afﬁne Hjelmslev Spaces with at least one cij not divisible by θ. Linear Code Constructions The number of codewords of C is

Fine Structure |C| = q|λ| = qλ1+···+λk . Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes Theorem and Geometry n over Finite Every linear code C ≤ RR is permutation equivalent to a linear Chain Rings code generated by a matrix of the following form: Thomas Honold Ik0 A01 A02 . . . A0,m−1 A0,m

Motivation 0 θIk1 θA12 ... θA1,m−1 θA1,m  0 0 2I 2A 2A  Finite Chain G := θ k2 ... θ 2,m−1 θ 2,m Rings ......  ......  Modules over  . . . .  Finite Chain  m−1 m−1   0 0 . . . 0 θ Ik θ Am−1,m Rings  m−1    Linear Codes Here ki ≥ 0 are integers satisfying k0 + k1 + · · · + km−1 ≤ n, Ik over Finite i Chain Rings denote ki × ki identity matrices over R, and with m−1 ki ×kj Projective and km := n − i=0 ki we have Aij ∈ R for all i < j. Afﬁne Hjelmslev Spaces Proof. P Suppose C has shape . Let k ′ ′ denote the Linear Code R λ i := λm−i − λm−i+1 Constructions number of parts of λ equal to m − i. Then Fine Structure k = rk(RC)= k0 + k1 + · · · + km−1. k×n Singer’s Arrange the basis c1,..., ck of RC as rows of a matrix G ∈ R Theorem (in that order). Then the ﬁrst k0 rows of G have height 0, the next Arcs and Blocking Sets k1 rows have height 1, etc.

Hyperovals Performing Gaussian elimination and permuting columns, if and Ovals necessary, we can transform G into the required form. Linear Codes Remarks and Geometry over Finite Chain Rings • In terms of k0, k1,..., km−1, the number of codewords of C is Thomas m k m−1 k k Pm−1(m−i)k Honold (q ) 0 (q ) 1 · · · q m−1 = q i=0 i .

Motivation • The submodules C := C[θi ] ∩ θi−1C of the Ulm-Kaplansky Finite Chain i Rings series of C are visible in G as follows: If we define Modules over Finite Chain Ik0 A01 A02 . . . A0,m−1 A0,m Rings 0 Ik1 A12 . . . A1,m−1 A1,m Linear Codes   ′ m−1 0 0 I A A over Finite G := θ k2 . . . 2,m−1 2,m , (⋆) Chain Rings  ......  Projective and  ......  Affine    0 0 . . . 0 Ik Am 1 m Hjelmslev  m−1 − ,  Spaces   ′ Linear Code then C1 = C[θ] is generated by G (which has k0 + · · · + km−1 Constructions ′ rows), C2 = C[θ] ∩ θC by the first k0 + · · · + km−2 rows of G , Fine Structure m−1 ′ . . . , and finally Cm = C[θ] ∩ θ C by the first k0 rows of G . Singer’s Theorem • Using the isomorphism θm−1R = Nm−1 =∼ R/N, we can Arcs and consider the Ci ’s as classical linear codes over Fq. (Simply Blocking Sets omit the factor θm−1 in (⋆) and read the rest modulo N.) This Hyperovals and Ovals viewpoint was adopted in [?, ?]. Linear Codes and Geometry over Finite Chain Rings Duality

Thomas Honold

Motivation n For x, y ∈ R write x · y := x1y1 + · · · + xnyn. Finite Chain Rings For S ⊆ Rn deﬁne the left (resp. right) dual code of S as Modules over Finite Chain ⊥ n Rings S := {x ∈ R ; x · y = 0 for all y ∈ S}, Linear Codes S⊥ := {x ∈ Rn; y · x = 0 for all y ∈ S}. over Finite Chain Rings

Projective and Afﬁne Remark ⊥ n Hjelmslev We have S ≤ RR regardless of whether S is (one-sided) linear Spaces or not, and similarly for S⊥. Linear Code Constructions Hence we can expect a suitable duality theory only if we deﬁne n ⊥ ⊥ Fine Structure the double dual of C ≤ RR as ( C) . Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes Theorem and Geometry n over Finite Let C ≤ RR be a left linear code over R of shape λ. Chain Rings ⊥ Thomas 1 The right linear code C has complementary shape Honold (m − λn, m − λn−1,..., m − λ1). ⊥ n Motivation In particular we have |C| · C = |R| , and C is free as an R-module iff C⊥ is free iff rk(C)+ rk(C⊥)= n. Finite Chain Rings 2 ⊥ C⊥ C Modules over ( )= Finite Chain ⊥ Rings 3 C 7→ C deﬁnes an antiisomorphism between the lattices of Linear Codes left resp. right linear codes of length n over R, and hence over Finite ′ ⊥ ⊥ ′⊥ ′ ⊥ ⊥ ′⊥ Chain Rings (C ∩ C ) = C + C , (C + C ) = C ∩ C .

Projective and Afﬁne Remark Hjelmslev Spaces Parts (2) and (3) are more generally true for quasi-Frobenius

Linear Code rings. Constructions

Fine Structure Proof. Since C ⊆ ⊥(C⊥), Part (2) follows from (1), and clearly implies Singer’s Theorem (3). Arcs and Blocking Sets (1) It is possible to derive this result working with standard

Hyperovals generator matrices, but the following is more conceptual and will and Ovals be needed later. Linear Codes and Geometry Proof cont’d. over Finite n Chain Rings Each y ∈ R induces a linear map RC → RR, x 7→ x · y. n Thomas In this way we obtain a homomorphism from RR to ♯ ⊥ Honold C = Hom(RC, RR)R with kernel C . Any linear map f C R can be extended to Rn (using, for Motivation : R → R R example, stacked bases for C and Rn) and hence has the form Finite Chain R R Rings x 7→ x · y. n ⊥ ∼ ♯ ⊥ ♯ Modules over This proves RR/C = C and that the shapes of C , C are Finite Chain Rings complementary. The proof of (1) is ﬁnished by the following

Linear Codes Lemma. over Finite Chain Rings Lemma Projective and ♯ Afﬁne A module R M and its dual M have the same shape (and in Hjelmslev particular the same number of elements). Spaces Linear Code Proof. Constructions r ∼ λi Fine Structure Write RM = i=1 R/N , and let x1,..., xr be a basis of RM. A linear map f : RM → RR is determined once f (x1),..., f (xr ) Singer’s L Theorem have been speciﬁed. For f (xi ) we can choose any ai ∈ R of λ m−λ Arcs and period dividing θ i , i.e. any ai ∈ N i . Blocking Sets r r ♯ ∼ m−λi ∼ λi Hyperovals This leads to M = i=1(N )R = i=1(R/N )R. and Ovals L L Linear Codes and Geometry over Finite Remark Chain Rings A self-dual linear code over R (i.e. C = C⊥) necessarily has Thomas self-complementary shape. This gives certain restrictions on the Honold parameters of C. Motivation

Finite Chain Examples Rings The linear codes C8 and O over Z4 generated by Modules over Finite Chain Rings 11111111 0022000001 Linear Codes 10002111 over Finite B00220000C 0 1 B C 01001231 Chain Rings B00022000C , B C B00101123C Projective and B00002200C B C Afﬁne B C @00011312A Hjelmslev B00000220C B C Spaces @00000022A Linear Code Constructions are both selfdual with shapes Fine Structure

Singer’s Theorem resp. Arcs and Blocking Sets The code O is the famous Octacode. Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Weight Spectra of Linear Codes For linear codes over a proper chain ring R the Hamming Thomas Honold distance dHam is not a good performance parameter. This is due to the following Motivation Finite Chain Fact Rings n For C ≤ R R we have dHam(C)= dHam C[θ] . Modules over Finite Chain So C cannot be better than the (usually much smaller) code C[θ] Rings (which is a classical linear code over R/N =∼ Fq). Linear Codes over Finite Chain Rings Proof. i Projective and Let c ∈ C with wHam(c)= dHam(C). If c has period θ , then i−1 i−1 i−1 Afﬁne θ c = (θ c1,...,θ cn) ∈ C[θ] \{0}. Hjelmslev i−1 Spaces Obviously wHam(θ c) ≤ wHam(c), so we must have equality and Linear Code dHam C[θ] = wHam(c)= dHam(C). Constructions Fine Structure Consequence Singer’s Theorem To produce good linear codes over chain rings, we should assign

Arcs and larger weights to those elements of R which generate small Blocking Sets ideals. Hence we must be able to distinguish between elements Hyperovals of R∗, N \ N2,..., Nm−1 \{0}, and {0}. and Ovals n Linear Codes For x = (x1,..., xn) ∈ R and 0 ≤ i ≤ m deﬁne and Geometry i i+1 over Finite ai (x) := {j; 1 ≤ j ≤ n and xj ∈ N \ N } . Chain Rings i Thomas i.e. ai (x) counts the entries of x which are “exactly” divisible by θ . Honold Deﬁnition Motivation n Finite Chain (i) The weight composition of x ∈ R is the (m + 1)-tuple of Rings integers a0(x), a1(x),..., am(x) . Modules over Finite Chain (ii) The weight enumerator of S Rn is the polynomial Rings ⊆

Linear Codes a0(x) am(x) over Finite AS(X0,..., Xm) := X0 · · · Xm ∈ C[X0,..., Xm]. Chain Rings x∈S Projective and X Afﬁne Hjelmslev Example Spaces The -linear code C 4 of shape generated by Linear Code Z4 ≤ Z4 Constructions

Fine Structure 1 1 1 1

Singer’s 0 2 2 0 Theorem 0 0 2 2 Arcs and   Blocking Sets has weight enumerator Hyperovals 4 4 2 2 4 and Ovals AC(X0, X1, X2)= 8X0 + X1 + 6X1 X2 + X2 . Linear Codes and Geometry over Finite Chain Rings Isomorphism of Linear Codes

Thomas Honold Deﬁnition Motivation Finite Chain n Rings (i) Two linear codes C1, C2 ≤ RR are said to be linearly n×n Modules over isomorphic if there exists a monomial matrix A ∈ R (a × Finite Chain matrix having exactly one nonzero entry aij ∈ R in each row Rings and column) such that C2 = {xA; x ∈ C1}. Linear Codes over Finite Chain Rings (ii) C1 and C2 are said to be semilinearly isomorphic if

Projective and C2 = {σ(x)A; x ∈ C1} with A as in (i) and σ ∈ Aut R. Afﬁne Hjelmslev Spaces

Linear Code Remark Constructions The monomial transformations Rn → Rn, x 7→ xA, are exactly n Fine Structure those automorphisms of the module RR which preserve the Singer’s weight composition. A similar remark applies to the Theorem “semimonomial” transformations of (ii). Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Multisets in PG(MR)

Thomas Deviating a little from our previous notation, we denote the lattice Honold of submodules of an arbitrary (not necessarily free) module MR by

Motivation PG(MR) and the set of nonzero cyclic submodules of PG(MR ) by . The elements of are called points. A point xR is degenerate Finite Chain P P Rings if xR is not free, i.e. |xR| < |R|. Modules over Finite Chain Definition Rings A multiset in PG(MR ) is a mapping K: P → N0. Linear Codes P over Finite The mapping K is extended to the power set 2 by defining Chain Rings Projective and K K Affine (Q)= (P) for Q⊆P. Hjelmslev P∈Q Spaces X Linear Code K Constructions For P = xR ∈P the integer (P) ≥ 0 is called the multiplicity of P in K. Fine Structure K K K Singer’s The integer (P)= P∈P (P) is called the cardinality of . Theorem The support of K is defined as supp K := P ∈P; K(P) > 0 , the Arcs and P Blocking Sets hull of K as the module hKi := Rx ≤ M , and the xR∈supp K R Hyperovals shape of K as the shape of hKi. and Ovals P Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Rings

Modules over Deﬁnition Finite Chain Two multisets K, K′ in PG(M ) resp. PG(M′ ) are said to be Rings R R equivalent if there exists a semilinear bijection φ: hKi → hK′i Linear Codes ′ over Finite satisfying K(P)= K φ(P) for every point P = xR ≤ hKi. Chain Rings Projective and Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Codes Associated to Multisets n Suppose C ≤ RR is any left linear code of length n over R. Thomas Honold Take any generator matrix G ∈ Rk×n of C (i.e. C = {xG; x ∈ Rk }, Motivation but the rows of G need not be independent). K k Finite Chain Write G = (g1| . . . |gn) and define a multiset in PG(RR) by Rings setting Modules over K Finite Chain (P) := |{j; 1 ≤ j ≤ n ∧ P = gj R}| Rings for P = gR ∈P. Linear Codes over Finite Definition Chain Rings We say that the multiset K and the code C are associated. Projective and Affine Hjelmslev Remarks Spaces

Linear Code • If C has full length (i.e. no universal zero coordinate), then Constructions K(P)= n. Fine Structure k Singer’s • One can restrict the definition to multisets in PHG(RR ) (i.e. Theorem allow only nondegenerate points) and so-called fat linear Arcs and codes. Blocking Sets Hyperovals • In general the relation C 7→ K is many-to-many. and Ovals Linear Codes and Geometry over Finite Chain Rings The Correspondence Theorem Thomas Theorem Honold K k For every multiset of cardinality n in PG(RR) there exists an n Motivation associated linear code C ≤ RR (which is necessarily of full K k1 K k2 Finite Chain length). Two multisets 1 in PG(RR ) and 2 in PG(RR ) Rings associated with full-length linear codes C1 and C2 over R, Modules over Finite Chain respectively, are equivalent if and only if the codes C1 and C2 are Rings semilinearly isomorphic. Linear Codes over Finite Chain Rings Proof. k Projective and The first part is easy: Choose a sequence g1,..., gn over R Affine such that the multiplicities in K of all points match those in the Hjelmslev n Spaces sequence g1R,..., gnR, and define C ≤ RR as the code Linear Code generated by (g1| . . . |gn). Constructions

Fine Structure The proof of the second part is quite technical and can be found

Singer’s in [?]. Theorem

Arcs and Remark Blocking Sets The theorem holds for a more general class of rings (including Hyperovals finite Frobenius rings). and Ovals Linear Codes and Geometry over Finite Chain Rings How to Compute the Parameters of C Thomas from Properties of K? Honold Parameters of C: length, shape (cardinality), weight enumerator Motivation The length of C equals K(P)). Finite Chain Rings For the shape use the following Modules over Proposition Finite Chain Rings K k A multiset in PG(RR) and an associated code C have the same Linear Codes shape. In particular, |hki| = |C|. over Finite Chain Rings Proof. Projective and k×n Affine Let G ∈ R be such that Hjelmslev Spaces C = {xG; x ∈ Rk } (Left row space of G) Linear Code Constructions hki = D := {Gy; y ∈ Rn} (Right column space of G) Fine Structure n Singer’s Define f : R → C, x 7→ xG. Theorem n n Arcs and C = Im f =∼ R / Ker f = R /⊥D =∼ D♯ (as left R-modules) Blocking Sets Hyperovals ♯ and Ovals Hence C, D and D all have the same shape. Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Remark Motivation The following example shows that the left row space and the left Finite Chain column space of a matrix G ∈ Rk×n usually have different shape. Rings

Modules over Finite Chain Example Rings Let a, b ∈ R such that ab 6= ba and Linear Codes over Finite 1 a Chain Rings G = . Projective and b ab Afﬁne Hjelmslev a Spaces 1 1 Left column space of G: R( b )+ R( ab )= R( b ) Linear Code Constructions Left row space of G: R(1, a)+ R(b, ab) ) R(1, a).

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes To compute the weight enumerator and Geometry over Finite i0 im Chain Rings AC (X0,..., Xm)= AiX0 · · · Xm i=(i ,...,i )∈I Thomas 0Xm Honold m+1 of C (where I consists of all m + 1-tuples i ∈ N0 satisfying Motivation i0 + · · · + im = n), we establish a correspondence between Finite Chain codewords of C and hyperplanes of PG(Rk ). Rings R k k Modules over All hyperplanes of PG(RR) (or PHG(RR)) have the form Finite Chain ⊥ k × Rings H = (Rx) for some x ∈ (R ) . Linear Codes Deﬁnition over Finite Chain Rings The K-type of H is the sequence (a0,..., am) where Projective and Afﬁne K Hjelmslev ai := (P) for 0 ≤ i ≤ m. Spaces P∈P X i k Linear Code P⊆H+θ R i 1 k Constructions P*H+θ + R Fine Structure

Singer’s Theorem Remark K k K Arcs and If is a multiset in PHG(RR) (i.e. supp contains only free Blocking Sets points), then ai counts the number of points of K which are Hyperovals and Ovals i-neighbours but not i + 1-neighbours to H. Linear Codes Proposition and Geometry over Finite Let K be a multiset in PG Rk associated with C Rn, and Chain Rings ( R ) ≤ R suppose the correspondence C 7→ K is given by G ∈ Rk×n. For Thomas ⊥ Honold each hyperplane H = (Rx) of K-type

Motivation (0,..., 0, aj ,..., am) with aj 6= 0 Finite Chain Rings the cyclic subcode RxG ≤ RC has weight enumerator Modules over Finite Chain Rings m−1 n m−s m−s−1 aj aj+1 aj+m−1−s aj+m−s+···+am Linear Codes Xm + (q − q )Xs Xs+1 · · · Xm−1 Xm over Finite Chain Rings Xs=j Projective and Afﬁne Proof. Hjelmslev Let G g g and c xG x g x g . Spaces = ( 1| . . . | n) = = ( · 1,..., · n) Linear Code Observation (easy to verify) Constructions i k i+1 k i i+1 Fine Structure gj ∈ H + θ R \ (H + θ R ) is equivalent to x · gj ∈ N \ N . Singer’s Hence (a0,..., am) is the weight composition of c. Theorem The rest follows from Rc =∼ Nj (as left R-modules) and the fact Arcs and Blocking Sets that multiplication by θ shifts the weight composition as follows:

Hyperovals and Ovals (b0,..., bm−1, bm) 7→ (0, b0,..., bm−2, bm−1 + bm). Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Remarks

Motivation • The weight enumerator of C can be computed from the Finite Chain Rings weight enumerators of the cyclic subcodes of RC using the Modules over principle of inclusion and exclusion. In the classical case this Finite Chain Rings is trivial, since C = {0} ⊎ c∈S Fqc \{0} . Linear Codes • Sometimes (especially in the case m 2) it is easier to over Finite U = Chain Rings compute the weight enumerators of C× and C \ C× Projective and separately. Afﬁne m m−1 × Hjelmslev If a0 6= 0, H gives rise to q − q codewords in C (the Spaces words in R×xG). All these have weight composition equal to Linear Code K Constructions the -type of H.

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes Example (Weight enumerator of the Octacode) and Geometry over Finite The Octacode O is generated by Chain Rings

Thomas Honold 10002111 01001231 Motivation 00101123 Finite Chain 00011312 Rings   Modules over   Finite Chain The associated multiset O is an arc in PHG(3, Z4) (i.e. no 4 points Rings of O are on the same hyperplane). Linear Codes over Finite Since 2O is the extended [8, 4, 4] Hamming code over {0, 2}, Chain Rings Projective and 8 4 4 8 Afﬁne A2O (X0, X1, X2)= X2 + 14X1 X2 + X1 . Hjelmslev Spaces Let ni be the number of planes of PHG(3, Z4) meeting O in i Linear Code Constructions points (0 ≤ i ≤ 3).

Fine Structure n0 + n1 + n2 + n3 = 120 Singer’s Theorem n1 + 2n2 + 3n3 = 8 · 28 Arcs and 8 Blocking Sets n2 + 3n3 = 2 · 6 8 Hyperovals n3 = 3 and Ovals Linear Codes and Geometry over Finite Chain Rings Example (cont’d) Thomas Honold Solving the system gives n0 = 8, n1 = n3 = 56, n2 = 0.

Motivation Since O/2O is the extended Hamming code, there is 1 O Finite Chain neighbourhood class of planes of -type (8, 0, 0) (the class ⊥ Rings containing Z4(1111) ) and 14 classes of O-type (4, ∗, ∗). Modules over Finite Chain O Rings -type #planes O Linear Codes (8, 0, 0) 8 H ∩ = ∅ over Finite (4, 1, 3) 56 |H ∩ O| = 3 Chain Rings (4, 3, 1) 56 |H ∩ O| = 1 Projective and Afﬁne Hjelmslev This gives Spaces 8 4 3 4 3 Linear Code AO× (X0, X1, X2)= 16X0 + 112X0 X1X2 + 112X0 X1 X2, Constructions 8 4 3 4 4 8 Fine Structure AO (X0, X1, X2)= X2 + 112X0 X1X2 + 14X1 X2 + 16X0 4 3 8 Singer’s + 112X0 X1 X2 + X1 Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings MacWilliams Identity

Thomas Theorem Honold n ⊥ n The weight enumerators of C ≤ RR and its dual code C ≤ RR Motivation are related by

Finite Chain Rings 1 AC⊥ (X0, X1,..., Xm)= · AC Am − Xm−1, Am−1 − qXm−2, Modules over |C| Finite Chain m−1 Rings ..., A1 − q X0, A0 , Linear Codes m−i m−i−1 over Finite where Ai = Xm + (q − 1)Xm−1 + · · · + (q − q )Xi is the Chain Rings weight enumerator of Ni . Projective and Afﬁne Sketch of proof. Hjelmslev Spaces (1) Deﬁne α ∈ Aut Q[X0,..., Xm] by Linear Code Constructions (αf )(X0, X1,..., Xm)= f Am − Xm 1, Am 1 − qXm 2, Fine Structure − − − m−1 Singer’s ..., A1 − q X0, A0 Theorem

Arcs and i Verify the identity A C A ⊥ for the m 1 codes N Blocking Sets α( C )= | | C + (0 ≤ i ≤ m) of length 1, whose weight enumerators A form a Hyperovals i and Ovals basis of hX0,..., XmiQ. Linear Codes Proof cont’d. and Geometry over Finite ⊥ (2) If C = C1 × C2 is a decomposable code and α(ACi )= |Ci | AC Chain Rings i ⊥ ⊥ ⊥ holds for i = 1, 2, then C = C × C , AC = AC AC , Thomas 1 2 1 2 Honold AC⊥ = A ⊥ A ⊥ and hence C1 C2

Motivation ⊥ ⊥ ⊥ α(AC )= α(AC1 AC2 )= α(AC1 )α(AC2 )= |C1| |C2| AC AC = |C| AC . Finite Chain 1 2 Rings n+m Modules over It follows that α(AC )= |C| AC⊥ also holds for the m Finite Chain completely decomposable codes Rings Linear Codes over Finite C = R ×···× R × N ×···× N ×···×{0}×···×{0}, Chain Rings n0 n1 nm Projective and Afﬁne | {z } | {z } Hjelmslev whose weight enumerators form a basis of Q|[X0,...,{zXm]n.} Spaces Linear Code (3) Show the existence of a Q-linear endomorphism of Constructions Q[X0,..., Xm] which sends AC to |C| AC⊥ . Fine Structure If χ is a generating character of R, then the Fourier transform Singer’s Theorem f y f x x y for f Rn n 0 Arcs and (F )( )= ( )χ( · ) ∈ Q ( ≥ ). Blocking Sets n xX∈R Hyperovals and Ovals has this property. Linear Codes and Geometry over Finite Chain Rings Axiomatic Deﬁnition Following HJELMSLEV,KLINGENBERG, and Thomas Honold KREUZER

Motivation Let Π = (P, L, I), I ⊆P×L, be an arbitrary (point-line) incidence Finite Chain Rings structure.

Modules over Finite Chain Definition (Neighbour relation) Rings Points x, y ∈P are said to be neighbours (notation: x ¨ y) if Linear Codes over Finite there exist distinct lines S, T incident with both x and y. Chain Rings Lines S, T ∈ L are said to be neighbours (notation: S ¨ T ) if Projective and there exist distinct points x, y incident with both S and T . (If lines Affine Hjelmslev are identified with subsets of P, this means just |S ∩ T |≥ 2.) Spaces Linear Code Further notation Constructions For x, y ∈P with x 6¨ y the symbol x, y denotes the unique line Fine Structure S ∈ L through x and y. Singer’s Theorem For x ∈P, S ∈ L the symbol x ¨ S denotes “there exists y ∈ S Arcs and with x ¨ y”. Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Deﬁnition (Projective Hjelmslev Space) Chain Rings An incidence structure Π = (P, L, I) with neighbour relation ¨ is Thomas Honold said to be a projective Hjelmslev space if it satisﬁes the following axioms: Motivation

Finite Chain (H1) For any two points x, y ∈P there exists a line S with Rings (x, S) ∈ I, (y, S) ∈ I. Modules over Finite Chain (H2) Every line S ∈ L contains at least three points which are Rings pairwise non-neighbours. Linear Codes over Finite (H3) For any x, y, z ∈P, x ¨ y and y ¨ z imply x ¨ z. Chain Rings

Projective and (H4) For any two lines S, T and any three points x, y, z with Afﬁne (x, S) ∈ I, (y, S) ∈ I, (x, T ) ∈ I, (z, T ) ∈ I, x 6¨ y, x 6¨ z, Hjelmslev Spaces y ¨ z, we have S ¨ T . Linear Code (H5) For a point x not incident with S ∈ L with x ¨ S, there Constructions always exist y, z ∈P with y 6¨ S, (z, S) ∈ I and (x, y, z) ∈ I. Fine Structure ¨ Singer’s (H6) Let x ∈P, S ∈ L with x 6 S and let y, z ∈ S. For every Theorem (y ′, x, y) ∈ I and every (z′, x, z) ∈ I there exists a line T with Arcs and (y ′, T ) ∈ I, (z′, T ) ∈ I and S ∩ T 6= ∅. Blocking Sets

Hyperovals and Ovals Linear Codes Assume that Π is a projective Hjelmslev space. and Geometry over Finite Remarks Chain Rings Thomas • Axiom (H2) allows us to identify lines with subsets of P. Honold • Π induces an incidence structure Π = (P, L, I), where Motivation P = [x]; x ∈P , L = [S]; S ∈ L denote the Finite Chain Rings corresponding partitions into equivalence classes and Modules over [x]I[S] : ⇐⇒ x ¨ S. The structure Π forms a classical Finite Chain Rings projective space.

Linear Codes over Finite Chain Rings Hjelmslev subspaces Projective and A subset T ⊆P is called a Hjelmslev subspace of Π if for every Afﬁne Hjelmslev two distinct points x, y ∈ T there exists a line L ∈ L with L ⊆ T Spaces and x, y ∈ L. Linear Code Constructions Hjelmslev subspaces form projective Hjelmslev spaces of their Fine Structure own in a natural way. Singer’s Theorem Subspaces

Arcs and For X ⊆P deﬁne the hull hX i as the intersection of all Hjelmslev Blocking Sets subspaces containing X . The set X is said to be a subspace if Hyperovals and Ovals X = hX i. Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation Independence Finite Chain Rings X ⊆P is said to be independent if for any x ∈ X we have ¨ Modules over x 6 hX \{x}i Finite Chain Rings Basis Linear Codes is said to be a basis of if and is independent. over Finite B⊆P Π hBi = P B Chain Rings

Projective and Dimension Afﬁne dim Π := |B| − 1 for any basis B of Π. Hjelmslev Spaces The equality dim Π= dim Π holds. Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Coordinate Hjelmslev Thomas Geometries Honold Suppose R is a ﬁnite chain ring and MR is a free right module ∼ k Motivation over R of rank k ≥ 3 (i.e. MR = RR). Finite Chain Rings Notation

Modules over PHG(MR) := (P, L, ⊆), where P and L denote the sets of all free Finite Chain rank 1, respectively free rank 2, submodules of M Rings R

Linear Codes The incidence structure PHG(MR) forms a projective Hjelmslev over Finite space (i.e., it satisﬁes Axioms (H1)–(H7)) Chain Rings ∼ ′ ∼ ′ Projective and MR = MR as R-modules implies PHG(MR) = PHG(MR ) as Afﬁne Hjelmslev incidence structures. Spaces

Linear Code A subset T ⊆P forms a Hjelmslev subspace of PHG(MR) iff T Constructions consists of all free rank 1 submodules of a free submodule U of Fine Structure MR. In this case we have dim T = rk(U) − 1. Singer’s Theorem Deﬁnition k Arcs and Π= PHG(RR) (which has geometric dimension dim Π= k − 1) is Blocking Sets called the (right) (k − 1)-dimensional projective Hjelmslev Hyperovals geometry over R. and Ovals Linear Codes and Geometry over Finite Chain Rings Coordinatization Theorem

Thomas Honold

Motivation

Finite Chain Rings

Modules over Finite Chain Theorem Rings

Linear Codes For every projective Hjelmslev space Π of dimension s ≥ 3, over Finite having on each line at least 5 points no two of which are Chain Rings

Projective and neighbours, there exists a chain ring R such that Afﬁne s+1 Hjelmslev PHG(RR ) is isomorphic to Π. Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Duality

Thomas Honold S(MR ) denotes the set of submodules of the R-module MR (which in case of M = Rk can be identified with the set of subspaces of Motivation k PHG(RR)). Finite Chain Rings k k Consider the map γ : S(RR ) → S(RR ), Modules over Finite Chain ⊥ k Rings U 7→ U := {y ∈ R ; x1y1 + · · · + xk yk = 0 for all x ∈ U}. Linear Codes over Finite Since U Rk is free of rank s iff U⊥ Rk is free of rank k s, Chain Rings ≤ R ≤ R − the map γ induces a correlation between the geometries Projective and k k Affine PHG(RR ) and PHG(RR). Hjelmslev Spaces In particular we can view hyperplanes (i.e. k − 2-dimensional Linear Code Hjelmslev subspaces) of the right coordinate geometry PHG Rk Constructions ( R ) as points of the left coordinate geometry PHG( Rk ) (and vice Fine Structure R versa). Singer’s Theorem 3 3 ∗ In the planar case (k = 3) we have PHG(R R )= PHG(R ) (dual Arcs and R Blocking Sets incidence structure). Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Affine Hjelmslev coordinate Thomas geometries Honold

Motivation

Finite Chain Rings k k AHG(RR) = (P, L, ∈), where P = R and Modules over k k × Finite Chain L = {a + uR; a ∈ R , u ∈ (R ) } Rings

Linear Codes In the planar case (k = 2) lines have equations over Finite Chain Rings y = mx + t (m, t ∈ R) resp. x = my + t (m ∈ N, t ∈ R) Projective and Afﬁne k Hjelmslev 2k 3(k−1) q −1 Spaces There are q points and q · q−1 lines Linear Code k+1 Constructions The incidence structure obtained by removing from PHG(RR ) a k Fine Structure neighbour class [H] of hyperplanes is isomorphic to AHG(RR ). Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Simplex Codes

Thomas Deﬁnition Honold K k The linear code C associated with the multiset in PG(RR), Motivation deﬁned by K(P)= 1 if P ∈P is a free point and K(P)= 0

Finite Chain otherwise, is called the k-dimensional simplex code over R and is Rings denoted by Sim(k, R). Modules over (k−1)(m−1) k Finite Chain The code Sim(k, R) has length q 1 q, shape (m,..., m), Rings k Linear Codes km over Finite and cardinality |Sim(k, R)| = q . Chain Rings | {z } All hyperplanes H of PG(Rk ) have the same K-type Projective and R Afﬁne (a0, a1,..., am) given by Hjelmslev Spaces (k−1)(m−1) k k − 1 (k−1)m Linear Code a0 = q − = q , Constructions 1 1 q q! Fine Structure (k−2)(m−1) k − 1 m−j m−j−1 Singer’s aj = q q − q , j = 1,..., m − 1, Theorem 1 q Arcs and Blocking Sets (k−2)(m−1) k − 1 am = q . Hyperovals 1 q and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

To prove this, observe that s j as is the number of free rank 1 Motivation ≥ submodules contained in H + θj Rk , a module of shape Finite Chain P Rings (m,..., m, m − j).

Modules over k 1 Finite Chain − Rings Hence| {z } Linear Codes over Finite (k−1)(m−1) k Chain Rings q 1 q if j = 0, as = Projective and q(k−1)(m−1) k−1 · q1−j if 1 ≤ j ≤ m. Afﬁne s≥j ( 1 q Hjelmslev X Spaces Solving for aj gives the stated formulas. Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Hamming Codes

Thomas Honold

Motivation

Finite Chain Rings Deﬁnition Modules over ⊥ Finite Chain The dual code Sim(k, R) is called the k-th order Hamming code Rings over R and is denoted by Ham(k, R). Linear Codes k over Finite Ham(k, R) is free of rank q(k−1)(m−1) − k. Chain Rings 1 q mq(k−1)(m−1)(k) −mk Projective and In particular |Ham(k, R)| = q 1 q . Afﬁne Hjelmslev Spaces The weight distribution of Ham(k, R) can be computed using the Linear Code MacWilliams identity. Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Thomas Example Honold ⊥ C := Sim(2, Z4) and C = Ham(2, Z4) are generated by Motivation Finite Chain 331000 Rings 101121 320100 Modules over , 011213 230010 Finite Chain Rings 310001 Linear Codes   over Finite   Chain Rings and have weight distributions

Projective and 6 4 2 4 Afﬁne AC(X0, X1, X2)= X2 + 3X1 X2 + 12X0 X1X2, Hjelmslev Spaces 1 A ⊥ (X , X , X )= A (X − X , X + X − 2X , X + X + 2X ) Linear Code C 0 1 2 16 C 2 1 1 2 0 1 2 0 Constructions

Fine Structure = · · ·

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Rings

Modules over Finite Chain Today’s Lecture: Projective Hjelmslev planes Rings over chain rings of length two Linear Codes over Finite Chain Rings

Projective and Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes k and Geometry Let Π = (P, L, I)= PHG(RR) be a coordinate Hjelmslev geometry over Finite ∼ Chain Rings over a finite chain ring R of length m with R/N = Fq. Thomas Definition (Refined neighbourhood relations) Honold Two points x = xR and y = yR are called i-neighbours Motivation i (0 ≤ i ≤ m, notation: x ï y) if |xR ∩ yR|≥ q . Finite Chain Two lines S and T are called i-neighbours (notation: S ¨ T ) if for Rings i every point x on S there exists a point y on T with x ¨ y. Modules over i Finite Chain Rings Remarks Linear Codes over Finite ¨ ¨ ¨ Chain Rings • 0 is the universal relation on P and L, 1= is the ordinary ¨ Projective and neighbour relation, and m is just equality. Affine i Hjelmslev • ï can be described as equality modulo N . Spaces Linear Code • ï defines equivalence relations on P and L. The Constructions corresponding classes are denoted by [x]i , [S]i . Fine Structure ¨ Singer’s • i is extended to Hjelmslev subspaces (of possibly different Theorem (i) (i) dimension) by ∆1 ï ∆2 iff π (∆1) ⊆ π (∆2) for the Arcs and corresponding images modulo Ni . Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Structure Theorems

Thomas Indispensable for doing Finite Geometry Honold (i) (i) Let P = [x]i ; x ∈P , L = [S]i ; S ∈ L . Motivation

Finite Chain Theorem Rings (i) (i) The incidence structure (P , L , ï ) is isomorphic to the Modules over projective Hjelmslev geometry PHG R Ni k . In particular, Finite Chain (( / )R/Ni ) Rings (1) (1) (1) (P , L , J ) is isomorphic to PG(k − 1, Fq). Linear Codes over Finite Chain Rings There are further theorems pertaining to the incidence structures Projective and induced on [x] resp. [S] (and, more generally, L-shaped Affine i i k Hjelmslev submodules of RR). For simplicity these will be stated only for the Spaces special case m = 2, k = 3. Linear Code Constructions Definition Fine Structure Nonempty intersections S ∩ [x] (where x ∈P, S ∈ L) are called Singer’s line segments of PHG R3 . Theorem ( R) The neighbour class T T x S x S is called the Arcs and ∈ L; ∩ [ ]= ∩ [ ] =[ ] Blocking Sets direction of the line segment S ∩ [x]. Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold Theorem 3 Motivation Let x be a point of PHG(RR). The incidence structure induced on Finite Chain [x] (with the line segments contained in [x] as lines) is isomorphic Rings to the affine plane AG(2, Fq ). Modules over Finite Chain Rings Theorem 3 Linear Codes Let S be a line of PHG(RR). The incidence structure induced on over Finite Chain Rings [S] (with the line segments having direction S as lines) is isomorphic to the dual affine plane (or punctured projective plane) Projective and ∗ Affine AG(2, Fq ) =∼ PG(2, Fq) \ p∞. Hjelmslev Spaces Example Linear Code Constructions Draw a picture for PHG(2, Z4). Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Singer’s Classical Theorem Thomas Theorem (Singer 1938) Honold PG(2, Fq) admits a cyclic collineation group G acting Motivation regularly (i.e. sharply transitive) on the points (and lines) of Finite Chain Rings PG(2, Fq). Modules over This leads to a simpliﬁed representation Finite Chain ′ ′ Rings PG(2, Fq) =∼ Dev(D) := (P , L , ∈) with Linear Codes over Finite ′ ′ G ∼ 2 D i i 2 Chain Rings P := = Zq +q+1, L := { + ; ∈ Zq +q+1},

Projective and Affine where D := {g ∈ G; g(p0) ∈ L0} for some fixed point-line Hjelmslev pair p L . Spaces ( 0, 0) ∈P×L Linear Code Proof. Constructions 3 Represent the ambient space as 3 (the cubic extension Fine Structure Fq Fq 3 Singer’s field of Fq). Choose a primitive element β of Fq and Theorem consider the collineation σ induced by Fq3 → Fq3 , x 7→ βx Arcs and 2 Blocking Sets (which has the same order q + q + 1 as the quotient group × × Hyperovals F 3 /Fq ). Let G = hσi. and Ovals q Linear Codes and Geometry over Finite Chain Rings Singer’s Theorem for Projective Thomas Hjelmslev Planes Honold Theorem (Hale-Jungnickel 1978) Motivation 3 Finite Chain The projective Hjlemslev plane PHG(RR), where R is either Rings Gq or Sq, admits a regular collineation group Modules over ∼ Finite Chain G = Zq2+q+1 × (Fq, +) × (Fq, +). Rings

Linear Codes Proof. over Finite Chain Rings R has a cubic Galois extension S (S = Gq3 if R = Gq, resp. ∼ 3 Projective and S = Sq3 if R = Sq). The ring S satisﬁes SR = RR. Afﬁne 3 Hjelmslev Hence we can view PHG(R ) as PHG(SR). Spaces R × Linear Code • xR ∈ P iff x ∈ S \ θS = S Constructions • xR yR iff y xR× Fine Structure = ∈ Singer’s This implies that Theorem Arcs and × × ∼ Blocking Sets G := S /R = Zq2+q+1 × (Fq, +) × (Fq, +) Hyperovals and Ovals acts regularly on P. Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation Finite Chain Example Rings Let G = Z7 × Z2 × Z2 = hai × hbi × hci. Modules over Finite Chain 3 3 Rings 1 If we set D = {1, b, a, ac, a b, a c}, then ∼ Linear Codes Dev(D) = PHG(2, Z4) and hai forms a hyperoval in Dev(D). over Finite 3 3 Chain Rings 2 If D = {1, b, a, ac, a , a bc}, then Dev(D) =∼ PHG(2, S2) and Projective and hai is a Baer subplane in Dev(D). Afﬁne Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Collineations 3 3 Every semilinear automorphism φ: RR → RR induces a Thomas 3 Honold collineation (i.e. an isomorphism) of PHG(RR) via φ(xR)= φ(x)R for xR ∈P. Motivation

Finite Chain Fundamental Theorem of Projective Hjelmslev Rings Geometry Modules over 3 Finite Chain Every collineation of PHG(RR) arises in this way, i.e. Rings 3 ∼ Aut PHG(RR) = PΓL(3, R). Linear Codes over Finite Projectivities of PHG 2 Chain Rings ( , Gq)

Projective and PGL(3, R)= GL(3, R)/Z(R) Affine Hjelmslev Notes Spaces k Linear Code • A corresponding theorem also holds for PHG(RR), k > 3. Constructions • Collineations preserve the partitions P, L of P resp. L into Fine Structure neighbour classes, as well as the sets of line segments resp. Singer’s Theorem dual line segments. Arcs and • PGL(3, R) acts regularly on ordered quadrangles of Blocking Sets PHG(R3 ), i.e. sets of 4 points which form a quadrangle in Hyperovals R and Ovals the quotient plane PG(2, Fq) Linear Codes 3 and Geometry General assumption: Π= PHG(RR), where R has length m = 2. over Finite Chain Rings Definition Definition Thomas K K Honold A multiset in (P, L, I) is called A multiset in (P, L, I) is called a (k, n)-arc (or simply an n-arc) a (k, n)-blocking multiset (or an Motivation if n-blocking multiset) if Finite Chain Rings (i) K(P)= k; (i) K(P)= k; Modules over Finite Chain (ii) K(L) ≤ n for every line (ii) K(L) ≥ n for every line Rings L ∈ L; L ∈ L; Linear Codes over Finite Chain Rings Further notes Projective and Affine K K K Hjelmslev • If is a set, then is an n-arc iff P\ is a Spaces (q2 + q − n)-blocking set. Hence projective arcs and blocking Linear Code sets are equivalent concepts. Constructions Fine Structure • We are interested in maximal arcs (i.e. (k, n)-arcs with the Singer’s largest possible k) or, more generally, complete arcs (i.e. Theorem (k, n)-arcs which cannot be extended to a (k + 1, n)-arc (and Arcs and Blocking Sets dually in minimal resp. irreducible blocking multisets).

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Maximal Arc Problem

Thomas Honold The maximal arc problem (for coordinate Hjelmslev planes) asks Motivation for the determination of the integers Finite Chain Rings 3 3 mn(RR)= max k; there exists a (k, n)-arc in PHG(RR) Modules over Finite Chain Rings for n ∈ N (or, less ambitious, only for 1 ≤ n ≤ q2 + q). Linear Codes over Finite Remarks Chain Rings Projective and • The maximal arc problem is inherently difficult and has been Affine Hjelmslev completely solved so far only for |R| = 4. (The completion of Spaces the case |R| = 9 is to be expected soon.) Linear Code Constructions • The corresponding problem in classical finite geometry is Fine Structure well-studied. Many results are available and can be used Singer’s along with the structure theorems for Π, [x], and [S]. Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The General Upper Bound

Thomas Honold Theorem Motivation K 3 Let be a (k, n)-arc in PHG(RR). Suppose there exists a Finite Chain K Rings neighbour class of points [x] with ([x]) = u and let ui ,

Modules over 1 ≤ i ≤ q + 1, be the maximum number of points on a line from Finite Chain the i-th parallel class in the afﬁne plane deﬁned on [x]. Then Rings

Linear Codes q+1 over Finite Chain Rings k ≤ q(q + 1)n − q ui + u. Projective and i=1 Afﬁne X Hjelmslev Spaces Linear Code Proof. Constructions For 1 ≤ i ≤ q + 1 let ℓi be a corresponding line segment in [x] of Fine Structure multiplicity ui . Count the sum of the multiplicities of the q(q + 1) Singer’s Theorem lines through the segments ℓi in two different ways, using the fact

Arcs and that these lines partition P\ [x]. Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Example Chain Rings We show that a 5-arc in PHG(2, Z9) or PHG(2, S3) can have at Thomas Honold most 40 points. K Motivation Case 1: has a 2-point, say x.

Finite Chain Each of the 12 lines through x can contain at most 3 further Rings points of K. =⇒ k ≤ 2 + 12 · 3 = 38 Modules over Finite Chain Case 2: 7 ≤ u ≤ 9 Rings K ∩ [x] must contain line segments in every direction. Each line Linear Codes through such a line segment can contain at most 2 further points over Finite Chain Rings from K outside [x]. =⇒ k ≤ u + 6 + 6 + 6 + 6 = 33 Projective and Case 3: 5 ≤ u ≤ 6 Afﬁne Hjelmslev K ∩ [x] must contain at least one line segment ℓ. Spaces =⇒ k ≤ u + 6 + 9 + 9 + 9 ≤ 39 Linear Code Constructions Case 4: u = 4 Fine Structure If the 4 points in K ∩ [x] form an oval (quadrangle), we get Singer’s k ≤ 4 + 4 · 9 = 40, otherwise Case 3 applies and gives k ≤ 37. Theorem Case 5: u ≤ 3 Arcs and Blocking Sets Here k ≤ 3 · 13 = 39. Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings

Thomas Honold

Motivation

Finite Chain Rings Remark Recently we have proved that m ( 3)= 39 and m ( 3)= 38. The Modules over 5 Z9 5 S3 Finite Chain corresponding maximal arcs have completely different structure. Rings (The first arc consists of 13 properly arranged triangles, one in Linear Codes over Finite each point class. The second arc has triangles in 9 point classes, Chain Rings which form an affine subplane of Π =∼ PG(2, F3). The remaining Projective and point classes contain 3, 2, 2 and 1 points. Affine Hjelmslev Spaces

Linear Code Constructions

Fine Structure

Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Tables Thomas Online tables of lower bounds (arising from constructions) are Honold maintained by the group in Bayreuth (KIERMAIER, KOHNERT): Motivation http://www.algorithm.uni-bayreuth.de/ Finite Chain Rings en/research/Coding_Theory/PHG_arc_table/index.html Modules over Finite Chain Rings 3 2 Table: Values of mn(RR ) for Hjelmslev planes of order q = 4 and Linear Codes 2 over Finite q = 9 Chain Rings 2 2 Projective and n/R Z4 F2[X]/(X ) Z9 F3[X]/(X ) Afﬁne 2 7 6 9 9 Hjelmslev Spaces 3 10 10 19 18 Linear Code 4 30 30 Constructions 5 39 38 Fine Structure 6 48 – 51 48 – 51 Singer’s Theorem 7 60 – 62 60 – 62

Arcs and 8 69 69 Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite 2 2 Chain Rings The Case q ≤ n ≤ q + q

Thomas Honold This case has been solved completely. Motivation It is more convenient to consider (k, n)-blocking multisets with Finite Chain 0 ≤ n ≤ q. Rings

Modules over In perfect analogy with the classical case we have Finite Chain Rings Theorem 3 Linear Codes The minimum size of a 1-blocking multiset in PHG(RR) is over Finite 2 2 Chain Rings k = q + q. A (q + q, 1)-blocking multiset is necessarily (the

Projective and characteristic function of) a line. Afﬁne Hjelmslev The existence part can be generalized to Spaces

Linear Code Theorem Constructions The minimum size of an n-blocking multiset, 1 ≤ n ≤ q, in Fine Structure 3 2 PHG(RR) is k = n(q + q). Singer’s Theorem Examples are provided by sums of n lines (which are not Arcs and projective if n ≥ 2). There exist projective examples as well. Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Classical Case

Thomas Honold In PG(2, Fq) the maximum size of a 2-arc (a set of points no three

Motivation of which are collinear) is given by

Finite Chain Rings r 3 q + 2 if q = 2 , Modules over m2(Fq )= Finite Chain (q + 1 if q is odd. Rings Linear Codes Examples of (q + 1, 2)-arcs (ovals) are provided by conics: over Finite Chain Rings K 2 2 Projective and = Fq(x, y, z); y − xz = 0 = (1, y, y ); y ∈ Fq ∪ (0, 0, 1) Afﬁne Hjelmslev r K Spaces In the case q = 2 the tangents to an oval are concurrent in the K Linear Code nucleus p (p = Fq(0, 1, 0) in the above example), so that ∪{p} Constructions is a (q + 2, 2)-arc (hyperoval). Fine Structure In the case q odd every oval is a conic (B. SEGRE’S THEOREM) Singer’s Theorem The tangents to an oval K form a dual (q + 1, 2)-arc. (Each point Arcs and outside K is on either 0 or 2 tangents.) Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings 2-Arcs in Hjelmslev planes 3 Thomas We keep the assumption Π= PHG(RR), where R has length Honold m = 2. Motivation Theorem Finite Chain 3 2 R q q 1 if R r (a Galois ring of characteristic 4). Rings m2( R)= + + = G2 2 Modules over The corresponding (q + q + 1, 2)-arcs have exactly one point in Finite Chain Rings each neighbour class and meet every line in 0 or 2 points

Linear Codes (hyperovals). over Finite Chain Rings Theorem 2 3 2 Projective and q + 2 ≤ m2(RR) ≤ q + q if R has characteristic 2. Afﬁne Hjelmslev Spaces Theorem 3 2 Linear Code m2(RR)= q if R has characteristic p > 2. Constructions The corresponding (q2, 2)-arcs have q2 neighbour classes Fine Structure containing 1 point and q + 1 empty neighbour classes which form Singer’s Theorem a line of the quotient plane PG(2, Fq). Arcs and Blocking Sets In the remaining case R = Gq, q odd, we have only the upper 3 2 Hyperovals bound m2(RR) ≤ q . and Ovals Linear Codes and Geometry over Finite Chain Rings The Upper Bounds

Thomas Let us show that Honold 2 Motivation 3 q + q + 1 if q is even, m2(RR ) ≤ 2 Finite Chain (q if q is odd. Rings

Modules over Finite Chain Rings Proof. If there exists a point class [x] with K([x]) ≥ 2, we get the stronger Linear Codes over Finite bound Chain Rings 3 2 m2(RR) ≤ 2 + 1 · 0 + q · q = q + 2. Projective and Afﬁne 3 2 Hjelmslev Otherwise of course m2(RR) ≤ q + q + 1. Spaces Now assume q is odd and consider the induced arc in a line Linear Code Constructions neighbour class [L] =∼ PG(2, Fq) \{p∞}. Fine Structure Since q is odd, at least one point class [x] on [L] must be empty. Singer’s =⇒ The set of empty point neighbour classes forms a blocking Theorem set in the quotient plane PG(2, Fq ). Hence there must be at least Arcs and Blocking Sets q + 1 empty classes (and if there are exactly q + 1, they must form a line of the quotient plane). Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings The Existence of Hyperovals in Thomas PHG(2, G2r ) Honold Idea of the proof 3 Motivation Represent PHG(RR) as PHG(SR ) where S = Gq3 is the cubic

Finite Chain Galois extension of R = Gq. Since Rings 6 3 Modules over × × q − q 2 2 Finite Chain Gq3 /Gq = 2 = q (q + q + 1), Rings q − q

Linear Codes over Finite there is a (unique) candidate set T of q2 + q + 1 points which Chain Rings × × forms a subgroup of G 3 /Gq . Projective and q Affine T Hjelmslev The set is invariant under a lifted Singer cycle Spaces =⇒ it suffices to test the points in one line class [L] for the 2-arc Linear Code property. Constructions Fine Structure Definition T 3 Singer’s is called Teichmüller set of PHG(RR ) Theorem T i 2 Arcs and This is due to the fact that = {Gqη ; 0 ≤ i ≤ q + q}, where η Blocking Sets 3 × has order q − 1 in Gq3 (Teichmüller or Hensel lift of a primitive Hyperovals and Ovals element β of Gq3 /pGq3 = Fq3 ). r Linear Codes For the actual computation one uses the following (with q = p ) and Geometry over Finite Fact Chain Rings Gq is isomorphic to the ring W2(Fq) of (truncated) Witt vectors of Thomas 2 Honold length 2 over Fq, which has underlying set Fq and operations

Motivation p−1 1 p Finite Chain j p−j (a0, a1) + (b0, b1) = (a0 + b0, a1 + b1 − a b ), Rings p j 0 0 j=1 Modules over X Finite Chain p p Rings (a0, a1) · (b0, b1) = (a0b0, a0b1 + b0 a1).

Linear Codes over Finite Chain Rings This reduces the arithmetic of Gq to that of Fq (in a way that only

Projective and depends on the characteristic p). For example, all Galois rings Affine r have the “same” operations Hjelmslev G2 Spaces a a b b a b a b a b Linear Code ( 0, 1) + ( 0, 1) = ( 0 + 0, 1 + 1 + 0 0), Constructions 2 2 (a0, a1) · (b0, b1) = (a0b0, a0b1 + b0a1). Fine Structure Singer’s With this isomorphism we have η = (β, 0) and, using {a} := (a, 0) Theorem p (Witt’s notation), p(a0, a1) = (0, a0), and thus Arcs and Blocking Sets 1/p Hyperovals (a0, a1) = (a0, 0) + (0, a1)= {a0} + p{a1 }. and Ovals Linear Codes and Geometry over Finite Chain Rings The existence of ovals in 3 Thomas PHG(R ), char R = p Honold R Idea of proof Motivation 2 Work in the affine Hjelmslev plane AHG(RR), representing the Finite Chain 2 Rings point set RR as a quadratic extension ring S of R. 2 Modules over It is rather obvious how to do this. If R = Fq[X; σ]/(X ) with Finite Chain i Rings p 2 σ(a)= a , take S = Fq[X; σ]/(X ) with the “same” σ. Linear Codes over Finite Points of AHG(SR) Chain Rings 4 p = a0 + a1X ∈ S, i.e. a0, a1 ∈ F 2 (q points) Projective and q Affine Hjelmslev Lines of AHG S Spaces ( R)

Linear Code L = a + uR = a0 + a1X + (u0 + u1X)(λ0 + λ1X); λ0, λ1 ∈ Fq , Constructions × where u0 6= 0 (since u ∈ S ) Fine Structure Point sets in AHG(SR) with exactly 1 point in each neighbour Singer’s Theorem class correspond with functions f : Fq2 → Fq2 via

Arcs and Blocking Sets pt = t + f (t)X

Hyperovals and Ovals Choose f as a σ-quadratic map. Linear Codes and Geometry over Finite 2 Chain Rings Quadratic maps of Fq

Thomas Honold

Motivation

Finite Chain Deﬁnition Rings 2 f : Fq2 → Fq2 is said to be σ-quadratic if f (λa)= σ(λ) f (a) for Modules over 2 2 2 2 Finite Chain a ∈ Fq , λ ∈ Fq, and if Fq × Fq → Fq , Rings (a, u) 7→ f (a + u) − f (a) − f (u) is a (symmetric) bilinear map. Linear Codes over Finite Chain Rings Theorem pi If σ(a)= a , the σ-quadratic maps of 2 are exactly the maps of Projective and Fq Afﬁne the form Hjelmslev Spaces 2pi 2pi q 2pi (q+1) f x Ax Bx Cx with A B C 2 Linear Code ( )= + + , , ∈ Fq , Constructions i Fine Structure i.e. those represented by a quadratic q-polynomial in x p . Singer’s Theorem

Arcs and Blocking Sets

Hyperovals and Ovals Linear Codes and Geometry over Finite Chain Rings Why Does This Work?

Thomas Observation Honold The point set K in AHG(SR ) corresponding to f : Fq2 → Fq2 forms × Motivation an oval iff for any a ∈ Fq2 , u ∈ Fq2 , λ ∈ Fq \{0, 1} the equation Finite Chain Rings pa+λu = pa + (pa+u − pa)(λ + µX) Modules over Finite Chain Rings has no solution µ ∈ Fq. Linear Codes For the X-component (only the X-component matters!) this over Finite Chain Rings translates into Projective and f (a + λu) − f (a) f (a + u) − f (a) µ Afﬁne − = ∈/ Fq. Hjelmslev σ(λ)u u σ(λ) Spaces

Linear Code Constructions Consequence Fine Structure K forms in oval in AHG(SR ), provided that f is σ-quadratic and Singer’s satisfies the condition Theorem Arcs and f (u) Blocking Sets × ∈/ Fq for all u ∈ F 2 . u q Hyperovals and Ovals Linear Codes and Geometry over Finite The Final Step Chain Rings Theorem Thomas There exist σ-quadratic maps satisfying this condition. Honold Proof. Motivation ∼ f can be considered as a map of PG (Fq2 )Fq = PG(1, Fq). For a Finite Chain f (u) Rings fixed point u the complementary condition is linear in f Fq u ∈ Fq Modules over and defines a hyperplane Hu of the projective space over Fq Finite Chain Rings formed by all σ-quadratic maps of Fq2 . This space is a PG(5, Fq). Linear Codes It can be shown that the q + 1 hyperplanes Hu do not cover all the over Finite Chain Rings points of this space. Equivalently they do not contain a common

Projective and solid (3-dimensional space). Afﬁne Hjelmslev Remarks Spaces

Linear Code • (9, 2)-arcs exist in the planes over both Z9 and S3. Constructions • The existence of (q2, 2)-arcs was proved soon after a Fine Structure computer search done earlier this year revealed the ﬁrst Singer’s Theorem (25, 2)-arc in the plane over Z25. 3 Arcs and • There is much evidence that m2(Z )= 21. If true we would Blocking Sets 25 have the ﬁrst example of chain rings R, S of the same order Hyperovals 3 3 and Ovals with char(R) < char(S) but mn(RR ) > mn(SS) for some n.