Linear Codes from the Axiomatic Viewpoint

Transcription

1 Linear Codes from the Axiomatic Viewpoint Jay A. Wood Department of Mathematics Western Michigan University jwood/ Noncommutative rings and their applications, IV University of Artois, Lens June 9, 2015

2 3. MacWilliams extension theorem and converse Extension property (EP) EP for Hamming weight over Frobenius bimodules via linear independence of characters Generalization for module alphabets Axiomatic viewpoint Parametrized codes and multiplicity functions Failure of EP for landscape matrix modules Converse of extension theorem: EP implies Frobenius () Axiomatic Viewpoint June 9, / 32

3 Notation Let R be a finite associative ring with 1. Let A be a finite unital left R-module: the alphabet. Let w : A Q be a weight: w(0) = 0. Extend to A n by n w(a 1,..., a n ) = w(a i ). i=1 () Axiomatic Viewpoint June 9, / 32

4 Symmetry groups Recall the symmetry groups of w: G lt = {u U(R) : w(ua) = w(a), a A}, G rt = {φ GL R (A) : w(aφ) = w(a), a A}. U(R) is the group of units of R, and GL R (A) is the group of invertible R-linear homomorphisms A A. Recall that I will usually write homomorphisms of left modules on the right side. () Axiomatic Viewpoint June 9, / 32

5 Monomial transformations Recall: if G GL R (A) be a subgroup, then a G-monomial transformation of A n is an invertible R-linear homomorphism T : A n A n of the form (a 1, a 2,..., a n )T = (a σ(1) φ 1, a σ(2) φ 2,..., a σ(n) φ n ), for (a 1, a 2,..., a n ) A n. Here, σ is a permutation of {1, 2,..., n} and φ i G for i = 1, 2,..., n. () Axiomatic Viewpoint June 9, / 32

6 Isometries Let C 1, C 2 A n be two linear codes. Recall that an R-linear isomorphism f : C 1 C 2 is a linear isometry with respect to w if w(xf ) = w(x) for all x C 1. Every G rt -monomial transformation is an isometry from A n to itself. () Axiomatic Viewpoint June 9, / 32

7 Extension property (EP) Given ring R, alphabet A, and weight w on A. The alphabet has the extension property (EP) with respect to w if the following holds: For any left linear codes C 1, C 2 A n, if f : C 1 C 2 is a linear isometry, then f extends to a G rt -monomial transformation A n A n. That is, there exists a G rt -monomial transformation T : A n A n such that xt = xf for all x C 1. () Axiomatic Viewpoint June 9, / 32

8 Slightly different point of view Linear codes are often presented by generator matrices. A generator matrix serves as a linear encoder from an information space to a message space. If f : C 1 C 2 is a linear isometry, then C 1 and C 2 are isomorphic as R-modules. Let M be a left R-module isomorphic to C 1 and C 2. Call M the information module. Then C 1 and C 2 are the images of R-linear homomorphisms Λ : M A n and N : M A n, respectively. Then, N = Λf : inputs on left! () Axiomatic Viewpoint June 9, / 32

9 Coordinate functionals C 1 was given by Λ : M A n. Write the individual components as Λ = (λ 1,..., λ n ), with λ i Hom R (M, A). Call the λ i coordinate functionals. Similarly, N = (ν 1,..., ν n ), ν i Hom R (M, A). The isometry f extends to a monomial transformation if there exists a permutation σ and φ i G rt such that ν i = λ σ(i) φ i for all i = 1,..., n. () Axiomatic Viewpoint June 9, / 32

10 Case of R Our first result will show that, for any finite ring R, A = R has EP with respect to the Hamming weight. It follows that A = R itself has EP with respect to the Hamming weight when R is Frobenius. The Frobenius ring case came first (Wood, 1999). The more general A = R case is due to Greferath, Nechaev, and Wisbauer (2004). () Axiomatic Viewpoint June 9, / 32

11 Techniques For any alphabet A, the summation formulas for characters imply that the Hamming weight wt satisfies wt(a) = 1 1 A π(a), a A. π Â Characters are linearly independent over C. () Axiomatic Viewpoint June 9, / 32

12 Symmetry groups for the Hamming weight Consider the Hamming weight wt on A = R, which is an (R, R)-bimodule. Both symmetry groups G lt and G rt equal U(R). () Axiomatic Viewpoint June 9, / 32

13 A preorder Define a preorder on Hom R (M, R) by λ ν if there exists r R such that λ = νr. It follows from a result of Bass that if λ ν and ν λ, then λ = νu, where u U(R). () Axiomatic Viewpoint June 9, / 32

14 Proof (a) R, A = R, with Hamming weight. C 1, C 2 R n, with f : C 1 C 2 linear isometry. R has a generating character: ρ : R C, ρ(π) = π(1) for π R. (Evaluate at 1 R.) Every π R has the form π = r ρ for some unique r R. C 1 is image of Λ : M R n ; C 2 is image of N : M R n. N = λf. Isometry: wt(xλ) = wt(xn), for all x M. () Axiomatic Viewpoint June 9, / 32

15 Proof (b) Hamming weight as character sum: n r ρ(xλ i ) = n s ρ(xν j ), x M. i=1 r R j=1 s R That is, n ρ(xλ i r) = i=1 r R n ρ(xν j s), x M. j=1 s R This is an equation of characters on M. () Axiomatic Viewpoint June 9, / 32

16 Proof (c) Among the λ i, ν j, choose one that is maximal for. Say, ν 1. Let j = 1 and s = 1 on the right side of the character equation. By linear independence of characters, there exists i and r R so that ρ(xλ i r) = ρ(xν 1 ) for all x M. Thus ρ(x(ν 1 λ i r)) = 1 for all x M. I.e., M(ν 1 λ i r) ker ρ. () Axiomatic Viewpoint June 9, / 32

17 Proof (d) By ρ a generating character, ν 1 = λ i r. Thus, ν 1 λ i. By maximality, ν 1 λ i and λ i ν 1. Thus, ν 1 = λ i u 1, for some u 1 U(R). Then inner sums agree: r R ρ(xλ ir) = s R ρ(xν 1s), x M. Set σ(1) = i. Subtract inner sums to reduce the size of the outer sums by 1. Proceed by induction. () Axiomatic Viewpoint June 9, / 32

18 Generalize to module alphabets For ring R, alphabet A, and Hamming weight wt, EP holds if A: (1) is pseudo-injective and (2) has a cyclic socle (embeds into R). Pseudo-injective means injective with respect to submodules. That is, if B is a submodule of A and h : B A is any module homomorphism, then h extends to h : A A. Main idea: use R-case to get GL R ( R)-monomial extension. Use pseudo-injectivity to show existence of GL R (A)-monomial extension. () Axiomatic Viewpoint June 9, / 32

19 Axiomatic viewpoint Assmus and Mattson, Error-correcting codes: an axiomatic approach, Consider linear codes up to monomial equivalence. What matters? Actually, I want to consider parametrized codes up to monomial equivalence. Usual set-up: ring R, alphabet A, weight w on A. A parametrized code is a finite left R-module M and an R-linear homomorphism Λ : M A n. () Axiomatic Viewpoint June 9, / 32

20 Scale classes The right symmetry group G rt acts on Hom R (M, A) on the right: λ λφ. Call the orbit space O = Hom R (M, A)/G rt. Denote orbit/ scale class of λ by [λ]. Up to G rt -monomial equivalence, a parametrized code Λ : M A n is completely determined by the number of coordinate functionals λ i belonging to the various classes [λ] O. () Axiomatic Viewpoint June 9, / 32

21 Multiplicity functions Let F (O, N) denote the set of functions η : O N. Call these multiplicity functions. Given a parametrized code Λ : M A n, define its multiplicity function η Λ by η Λ ([λ]) = {i : λ i [λ]}. Other authors: multisets, value function (Chen, et al.), projective systems, etc. No zero columns: F 0 (O, N) = {η : η([0]) = 0}. () Axiomatic Viewpoint June 9, / 32

22 Weights of elements Given Λ : M A n, consider the weights w(xλ) for x M. The weights w(xλ), x M, depend only on η Λ, not Λ itself: G rt -monomial transformations are isometries. In fact: w(xλ) = w(xλ)η Λ ([λ]), x M. [λ] O () Axiomatic Viewpoint June 9, / 32

23 Invariance under G lt If u G lt, then w((ux)λ) = w(u(xλ)) = w(xλ), for all x M. G lt acts on M on the left: x ux, x M. Denote orbit space by O = G lt \M. w(0λ) = w(0) = 0. Denote F 0 (O, Q) = {f : O Q, f (0) = 0}. () Axiomatic Viewpoint June 9, / 32

24 Well-defined W map We get a well-defined map W : F 0 (O, N) F 0 (O, Q), with W (η)(x) = w(xλ)η([λ]), [λ] O for x O, η F 0 (O, N). () Axiomatic Viewpoint June 9, / 32

25 Completion over Q F 0 (O, N) is an additive semi-group, and F 0 (O, Q) is a Q-vector space. The map W is additive. The addition in F 0 (O, N) corresponds to concatenation of generator matrices. By tensoring over Q, we get a Q-linear transformation of Q-vector spaces: W : F 0 (O, Q) F 0 (O, Q). () Axiomatic Viewpoint June 9, / 32

26 Re-interpretation of EP An alphabet A has EP with respect to a Q-valued weight w if and only if the linear map W : F 0 (O, Q) F 0 (O, Q) is injective for all information modules M. Bogart, et al., Greferath, () Axiomatic Viewpoint June 9, / 32

27 Matrix modules and Hamming weight What does W look like for matrix module alphabets? Let R = M k k (F q ), A = M k l (F q ), with Hamming weight wt. Symmetry groups: G lt = U(R) = GL(k, F q ); G rt = GL R (A) = GL(l, F q ). () Axiomatic Viewpoint June 9, / 32

28 Orbit spaces For M = M k m (F q ), Hom R (M, A) = M m l (F q ). Then O = G lt \M = GL(k, F q )\M k m (F q ), which is the set of row reduced echelon (RRE) matrices of size k m. And O = Hom R (M, A)/G rt = M m l (F q )/GL(l, F q ), which is the set of column reduced echelon (CRE) matrices of size m l. () Axiomatic Viewpoint June 9, / 32

29 Dimension counting First note that dim Q F 0 (O, Q) = O 1 and dim Q F 0 (O, Q) = O 1. So, dim Q F 0 (O, Q) is the number of nonzero RRE matrices of size k m. And dim Q F 0 (O, Q) is the number of nonzero CRE matrices of size m l. If k < l and k < m, there are more of the CRE matrices than the RRE matrices; i.e., dim Q F 0 (O, Q) > dim Q F 0 (O, Q). This says that EP fails when k < l. ( Landscape ) () Axiomatic Viewpoint June 9, / 32

30 Converse of EP for Hamming weight We claim: if an alphabet A has EP for the Hamming weight, then A (1) is pseudo-injective and (2) has a cyclic socle. Likewise: if a ring R has EP for the Hamming weight, then R is Frobenius (which means Soc(R) is cyclic). We follow a strategy of Dinh and López-Permouth, () Axiomatic Viewpoint June 9, / 32

31 Proof If Soc(A) is not cyclic (same idea for R), then Soc(A) contains a matrix module of the form A = M k l (F q ) with k < l. There exist counter-examples to EP over A. Regard these codes as codes over A: A Soc(A) A. They are also counter-examples over A. Pseudo-injectivity is equivalent to the length 1 case of EP (Dinh, López-Permouth). () Axiomatic Viewpoint June 9, / 32

32 Other uses of W map We will see the W map again. Other weight functions. Linear one-weight codes. Isometries of additive codes. () Axiomatic Viewpoint June 9, / 32