Automorphisms of Additive Codes

Transcription

1 Automorphisms of Additive Codes Jay A. Wood Western Michigan University jwood 32nd Ohio State-Denison Mathematics Conference Columbus, Ohio May 9, 2014

2 Acknowledgments/the Question I thank Philippe Langevin for the following question: Let K be a subfield of a finite field L. Like in the classical case, we see that coordinate permutations and component-wise K-linear isomorphisms of L preserve the Hamming weight of L n. Now let C be a K-subspace of L n, and let f be a K-linear isomorphism of C that preserves the Hamming weight. I wonder if it is true that f extends as a map like above? (by , May 9, 2013) JW (WMU) Automorphisms of Additive Codes May 9, / 22

3 Short Answer No. JW (WMU) Automorphisms of Additive Codes May 9, / 22

4 Definitions Let K be a finite field, and let L be a finite dimensional vector space over K. A K-linear code over L of length n is a K-linear subspace C L n. We use the Hamming weight on L. This is a crucial hypothesis. The Hamming weight on L differs from the Hamming weight on K l, where l = dim K L. This can be generalized to finite rings K and finite module alphabets L. JW (WMU) Automorphisms of Additive Codes May 9, / 22

5 Additive Codes Let L = F q, q = p l, and K = F p. Then K-linear codes over L are additive codes over L. Such codes are closed under addition. It follows that they are closed under K-scalar multiplication. Monomial transformations: permutations and component-wise application of K-linear isomorphisms of L (not field automorphisms). Monomial transformations preserve the Hamming weight coming from L. JW (WMU) Automorphisms of Additive Codes May 9, / 22

6 Generator Matrix A K-linear code is often given by a generator matrix G. The rows of G form a K-basis for C L n. If G has size m n, then G defines an injective K-linear map Λ : M = K m L n, whose image is the K-linear code C. (Inputs on the left.) JW (WMU) Automorphisms of Additive Codes May 9, / 22

7 Isometries An isometry of a K-linear code C L n is an invertible K-linear map f : C C that preserves the Hamming weight. In terms of Λ : M L n : an element f GL K (M) such that wt(xf Λ) = wt(xλ) for all x M. All the isometries of C form a group, the isometry group Isom(C) GL K (M). JW (WMU) Automorphisms of Additive Codes May 9, / 22

8 Monomial Transformations A monomial transformation of L n is an invertible K-linear transformation T : L n L n of the form (a 1,..., a n )T = (a σ(1) φ 1,..., a σ(n) φ n ), where σ is a permutation of {1,..., n} and the φ i are invertible K-linear transformations of L. Define the monomial group Monom(C) = {T monomial on L n : CT = C}. JW (WMU) Automorphisms of Additive Codes May 9, / 22

9 Restriction Map Monomial transformations preserve weight. By restricting to C, we have a natural map restr : Monom(C) Isom(C). Langevin s question: is this map onto? In matrix terms, if f Isom(C), is there a T Monom(C) such that fg = GT? If L = K, then the restriction map is always onto. (MacWilliams, 1961) JW (WMU) Automorphisms of Additive Codes May 9, / 22

10 Example Let K = F 2, L = F 4 = F 2 [ω]/(ω 2 + ω + 1). Let C L 3 be the additive code generated by: G = 1 ω 0 ω What are restr(monom(c)) and Isom(C)? JW (WMU) Automorphisms of Additive Codes May 9, / 22

11 Monomial Group restr(monom(c)) is a Klein 4-group, generated by f 1, f 2, below. f 1 = f 2 = e.g. f 1 f 2 G = 1 ω 0 ω 2 ω ω 1 = G 0 ω 0 ω JW (WMU) Automorphisms of Additive Codes May 9, / 22

12 Isometry Group However, Isom(C) is a dihedral group of order 8, generators f 1 and f 3 (below): f1 2 = 1, f 3 4 = 1, f 1 f 3 f 1 = f 1 3. f 3 = JW (WMU) Automorphisms of Additive Codes May 9, / 22

13 Weight Preservation List of codewords and their images under f 3 : ω 0 ω 1 0 ω 2 ω ω 1 ω ω ω ω ω 1 ω 2 ω 2 0 ω 1 0 ω ω 2 1 ω JW (WMU) Automorphisms of Additive Codes May 9, / 22

14 Non-Extendability of f 3 Compare G and f 3 G: G = 1 ω 0 ω 1 0 f 3 G = 1 ω ω 2 ω 2 0 The patterns of the columns are not compatible via a monomial transformation. JW (WMU) Automorphisms of Additive Codes May 9, / 22

15 How Bad Can Things Get? Generalize to linear codes over matrix modules: K = M k k (F q ), L = M k l (F q ), M = M k m (F q ). Theorem Assume k < l < m. Pick any two subgroups G 1 G 2 GL K (M) (subject to a closure condition). Then there exists a K-linear code C L n with underlying module M such that Isom(C) = G 2 and restr(monom(c)) = G 1. (Length n could be very big.) JW (WMU) Automorphisms of Additive Codes May 9, / 22

16 Linear Codes via Functionals (a) Assmus-Mattson (1963). Let K be a finite ring and L, M be finite left K-modules; L is the alphabet, and M is the information space. Given functionals λ 1,..., λ n M := Hom K (M, L), the image of Λ : M L n, x (xλ,..., xλ n ), is a K-linear code in L n. If one fixes a set of generators x 1,..., x k for M, then the matrix with (i, j)-entry x i λ j is a generator matrix for this K-linear code. JW (WMU) Automorphisms of Additive Codes May 9, / 22

17 Linear Codes via Functionals (b) G = GL K (L), the group of K-linear isomorphisms of L, acts on M = Hom K (M, L). Up to monomial equivalence, all that matters is the number of times the G-orbit of a functional λ M appears in the list λ 1,..., λ n : call this number the multiplicity η(λ). Conversely, a multiplicity function η : M G N determines a linear code, up to monomial equivalence. JW (WMU) Automorphisms of Additive Codes May 9, / 22

18 Linear Codes via Functionals (c) For the Hamming weight wt on L, the weight of a codeword is wt(x) = λ M G wt(xλ) η(λ), x M. This defines an additive map of function spaces W : F (M G, N) F (M, N). For linear codes over matrix modules and the Hamming weight, W has a nonzero kernel exactly when k < l. JW (WMU) Automorphisms of Additive Codes May 9, / 22

19 Linear Codes via Functionals (d) Tensor over Q to get a Q-linear transformation W : F (M G, Q) F (M, Q). For linear codes over matrix modules and the Hamming weight, if k < l, then it is possible to write down an explicit basis for ker W. JW (WMU) Automorphisms of Additive Codes May 9, / 22

20 Action by GL K (M) The K-linear isomorphisms GL K (M) act on M = Hom K (M, L), on M G, and on F (M G, Q). A linear code C with underlying module M corresponds, up to monomial equivalence, to a multiplicity function η F (M G, Q). Let f GL K (M). Then f restr(monom(c)) when f η = η, and f Isom(C) when f η η ker W. JW (WMU) Automorphisms of Additive Codes May 9, / 22

21 Sketch of Proof of Theorem By averaging if necessary, find η F (M G, Q) that is invariant under the group G 2 but not invariant under any larger subgroup. (This is where the closure condition plays a role.) So far, restr(monom(c)) = Isom(C) = G 2. By using the explicit basis of ker W, modify η to obtain η such that η η ker W, but η is invariant only under G 1 (and no larger). Then restr(monom(c )) = G 1 and Isom(C ) = Isom(C) = G 2. JW (WMU) Automorphisms of Additive Codes May 9, / 22

22 Corollaries Can choose G 1 = F q id M (minimal) and G 2 = GL K (M) (maximal). Such an example over K = F 2 and L = F 4 has length n = 24. Examples can then be found over any non-frobenius ring R, since the socle contains a matrix module with k < l. JW (WMU) Automorphisms of Additive Codes May 9, / 22