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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.
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