Symmetries of a q-ary Hamming Code

Transcription

1 Symmetries of a q-ary Hamming Code Evgeny V. Gorkunov Novosibirsk State University <evgumin@gmail.com> Algebraic and Combinatorial Coding Theory Akademgorodok, Novosibirsk, Russia September 5 11, 2010

2 Notation F q = GF(q) the Galois field of order q = p r F n q the n-dimensional vector space over F q d(x, y) = #{i : x i y i } the Hamming distance w(x) = #{i : x i 0} weight of x F n q supp(x) = {i : x i 0} the support of x F n q C F n q a q-ary code of length n; d(c) = min{d(x, y): x, y C, x y} the minimum distance of C

3 Definitions Code equivalence Two codes are equivalent if there is an isometry of F n q that maps one of the codes into the other one

4 Definitions Perfect codes The balls with radius 1 centred at the codewords partition the space F n q Such codes have the minimum distance d = 3 Golay codes Binary and ternary Golay codes and codes equivalent to them have d = 7 and d = 5

5 Definitions Perfect codes The balls with radius 1 centred at the codewords partition the space F n q Such codes have the minimum distance d = 3 Golay codes Binary and ternary Golay codes and codes equivalent to them have d = 7 and d = 5

6 Structure investigations Faces of regularity Linearity Rank of a code Dimension of the kernal of a code Perfect and uniformly packed codes Distance invariance Complete regularity Regular properties of minimal distance graph Other extremal properties

7 Structure investigations Automorphisms and symmetries Automorphism group of a code C The group of isometries of the space F n q that map the code C into itself Symmetry group of a code C The group of automorphisms of C that fix the vector 0.

8 Isometries and automorphisms Examples Permutation π S n a permutation on coordinate positions, n = 3 : (x 1, x 2, x 3 )(123) = (x 3, x 1, x 2 ) Configuration σ = (σ 1,..., σ n ) S n q n permutations on elements of F q, n = 3 : (x 1, x 2, x 3 )σ = (x 1 σ 1, x 2 σ 2, x 3 σ 3 )

9 Isometries and automorphisms Isometries of the space F n q Theorem (Markov, 1956) The group of isometries of the space F n q is Aut(F n q) = S n Sq n = {(π; σ): π S n, σ Sq n } with multiplication given by (π; σ)(τ; δ) = (πτ; στ δ)

10 Isometries and automorphisms Kinds of isometries: q = 2 Permutation automorphisms π S n PAut(F 2 ) = Sym(F 2 ) All configurations are translations S 2 acts on F 2 : e β + 0 (0 1) β + 1 x F 2, σ S n 2 xσ = x + v for some v F 2

11 Isometries and automorphisms Kinds of isometries: q = 3 Permutation automorphisms π S n PAut(F 3 ) Sym(F 3 ) Monomial configurations S 3 acts on F 3 : e 1 β (1 2) 2β x F 3, σ multiplying xσ = xd for some diagonal matrix D Monomial automorphisms x F 3, π S n, σ multiplying x(π; σ) = xpd = xm MAut(F 3 ) = Sym(3) for some monomial matrix M

12 Isometries and automorphisms Kinds of isometries: q 4 Permutation automorphisms π S n PAut(F n q) Sym(F n q) Configurations q = 4 Gal(F 4 ),, + (0 α 2 1 α) (β + α) 2, where α primitive element of F 4 no matrix representation for all! q 5 no field operations for all!

13 Linear and semilinear transformations of F n q General linear group GL n (q) f (αx + βy) = αf (x) + βf (y) for all x, y F n q and α, β F q General semilinear group ΓL n (q) = Gal(F q ) GL n (q) f (αx + βy) = γ(α)f (x) + γ(β)f (y) for all x, y F n q, all α, β F q, and some γ Gal(F q ) But not all of them are isometries of F n q!

14 MacWilliams theorem Theorem (MacWilliams, 1962) Two linear codes are monomially equivalent iff there exists an isomorphism between them (as linear spaces) preserving the weight of each vector Corollary 1 MAut(F n q) all linear symmetries of F n q Corollary 2 Gal(F q ) MAut(F n q) all semilinear symmetries of F n q

15 The automorphism group of a linear code Proposition If a code C F n q is linear, then Aut(C) = Sym(C) C Theorem C is an [n, n m, d 3] q -code The semilinear symmetry group of C is isomorphic to some subgroup of ΓL m (q)

16 What is known Semilinear automorphisms of H Theorem The semilinear symmetry group of a q-ary Hamming code H of length n = qm 1 q 1 is isomorphic to ΓL m(q) q = 2, 3 all symmetries of F n q are linear Aut(H) = GL m (q) H q 4 not all symmetries of F n q are semilinear Aut(H) = ΓL m (q) H?

17 What is known Is there anything to doubt? Example C F n q is the linear code with H = [ ] A S q is a linear transformation of F q as a vector space over the subfield F p (e, (A, A,..., A)) Aut(C) for q 8 there exists A such that this automorphism of C is neither linear nor semilinear

18 Results Collinear triples T = {x H: w(x) = 3} Sym(H) Sym(T ) Lemma x, y T supp(x) = supp(y) y = µx for some µ F q

19 Results Symmetries of Hamming triples Lemma (saving collinearity) (π; σ) Sym(T ) (π; σ) preserves collinearity of vectors from F n q Lemma (saving sum) (π; σ) Sym(T ) (π; σ) preserves sum of vectors from F n q Lemma (π; σ) Sym(T ) (π; σ) is a semilinear transformation of F n q

20 Results The automorphism group of H Theorem For any q-ary Hamming code Hof length n = qm 1 q 1, where q, m 2, it is true Aut(H) = ΓL m (q) H

21 Conclusion We proved that all symmetries of the Hamming code are semilinear the same can be said about the triple system of a q-ary Hamming code