Foundational Aspects of Linear Codes: 3. Extension property: sufficient conditions

Transcription

1 Foundational Aspects of Linear Codes: 3. Extension property: sufficient conditions Jay A. Wood Department of Mathematics Western Michigan University jwood/ On the Algebraic and Geometric Classifications of Projective Varieties University of Messina June 22, 2016

2 3. Extension property: sufficient conditions Extension property (EP) EP for Hamming weight over Frobenius bimodules via linear independence of characters Egalitarian weights for module alphabets General weight: reducing to egalitarian weight Weights with maximal symmetry Lee and Euclidean weights on Z/NZ Foundational Aspects June 22, / 37

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 C be a weight: w(0) = 0. Extend to A n by n w(a 1,..., a n ) = w(a i ). i=1 Hamming weight: wt(0) = 0; wt(a) = 1 for a 0. Foundational Aspects June 22, / 37

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. Hamming weight: G lt = U(R), G rt = GL R (A). Recall that I will usually write homomorphisms of left modules on the right side. Foundational Aspects June 22, / 37

5 Monomial transformations Recall: if G GL R (A) is 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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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! Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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 (JW, 1999). The more general A = R case is due to Greferath, Nechaev, and Wisbauer (2004). Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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). Foundational Aspects June 22, / 37

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). Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

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. Foundational Aspects June 22, / 37

19 Using generating character to define a weight Suppose the alphabet A admits a generating character ρ: Soc(A) is cyclic. Fix a subgroup U GL R (A). Define a weight w U : A C: w U (a) = 1 1 U ρ(aφ), a A. φ U When A = R, the ring must be Frobenius. Foundational Aspects June 22, / 37

20 Properties of w U w U (0) = 0. Re-index: U G rt (w U ). By the summation formulas and ρ generating: for any nonzero left R-submodule B A, and any a 0 A, w U (a 0 + b) = B. b B We say that w U is egalitarian on cosets of B. Foundational Aspects June 22, / 37

21 A little history w U is similar a formula of Honold s for the homogeneous weight. Homogeneous weight was developed first for Z/mZ, then for more general rings and modules. Constantinescu, Heise, Greferath, Schmidt, Honold, Nechaev. The following proof is due essentially to Barra and Gluesing-Luerssen (2014). Foundational Aspects June 22, / 37

22 w U has EP Suppose w U (xλ) = w U (xn) for all x M. Equation of characters: for all x M, n ρ(xλ i φ) = i=1 φ U n ρ(xν j ψ). j=1 ψ U Use linear independence of characters: for j = 1, ψ = id A, there exist i = σ(1) and φ 1 U with ρ(xλ σ(1) φ 1 ) = ρ(xν 1 ) for all x M. ρ generating: ν 1 = λ σ(1) φ 1. Inner sums agree, reduce outer sum, and continue by induction. Foundational Aspects June 22, / 37

23 Correlation action (Greferath-Honold) Let α : R C and w : A C. Define the correlation wα : A C by (wα)(a) = w(ra)α(r), a A. r R If w(0) = 0, then (wα)(0) = 0. G rt (w) G rt (wα). Foundational Aspects June 22, / 37

24 Comparing isometries Lemma If f : C A n is a w-isometry, then f is a wα-isometry for any α. Proof. For x C, (wα)(xf ) = r R w(rxf )α(r) = r R w(rx)α(r) = (wα)(x). Foundational Aspects June 22, / 37

25 A sufficient condition for EP Theorem Let U = G rt (w). If wα = w U for some α, then w has EP. Proof. Suppose f : C A n is a w-isometry. By the previous lemma, f is a w U -isometry. Since w U has EP, f extends to a U-monomial transformation. Thus w has EP. Foundational Aspects June 22, / 37

26 Matrix representing correlation Recall that (wα)(a) = r R w(ra)α(r), for a A. By symmetry, values of w(ra) depend only on orbit [r] of r under G lt and orbit [a] of a under G rt. Define a complex matrix W = (w(ra)), [0] [r] G lt \R, [0] [a] A/G rt. w has EP when rk W = A/G rt 1. When W is square (e.g., when A = R is commutative), w has EP when W is invertible or det W 0. Foundational Aspects June 22, / 37

27 Cases of maximal symmetry There has been progress on finding more explicit conditions over ring alphabets (A = R) when the weight w has maximal symmetry: G lt = G rt = U(R). When R is a product of chain rings: Greferath, Mc Fadden, Zumbrägel, When R is a principal ideal ring: Greferath, Honold, Mc Fadden, Wood, Zumbrägel, Here det W is factored into terms 0<dR ar w(d)µ(0, dr), for a R, where µ is the Möbius function for the poset of principal right ideals of R. Foundational Aspects June 22, / 37

28 Case of commutative chain rings A finite ring is a chain ring if all its left ideals form a chain under inclusion. In that case, every ideal is two-sided. A chain ring is Frobenius. In the commutative case, a chain ring is a local principal ideal ring. Examples: fields; Z/p k Z, p prime; Galois rings (Galois extensions of Z/p k Z). Foundational Aspects June 22, / 37

29 det W factors over chain rings Let R be a finite commutative chain ring with weight w. Let U be the symmetry group of w. When R is a finite field F, det W factors as a product of Fourier transforms of w with respect to characters of the quotient group G = F /U. This is due to Dedekind and Frobenius, This generalizes for finite commutative chain rings, 1998: det W factors as a product of character sums of w over U(R)/U that reflect the conductors of the characters. Foundational Aspects June 22, / 37

30 Examples over Z/NZ In addition to the Hamming weight, there are three additional weights that are easy to define on Z/NZ. Lee weight: viewing Z/NZ = {0, 1,..., N 1}, Lee weight is w L (a) = min{a, N a}. Euclidean weight: w E (a) = w L (a) 2. Complex Euclidean weight: w(a) = exp(2πia/n) 1 2 = 2 2 cos(2πa/n), the square of distance to 1 in C. Foundational Aspects June 22, / 37

31 Facts about EP over Z/NZ The complex Euclidean weight is easy: it is the egalitarian weight using U = {±1}. EP for w L and w E have been numerically verified (W invertible via SVD) for N Proofs of EP for w L and w E using the form of cyclotomic polynomials for N = 2 k, 3 k and N = p = 2q + 1 (Langevin-JW) or p = 4q + 1 (Barra) where p and q are prime. Foundational Aspects June 22, / 37

32 EP over Z/p k Z Recent work of Dyshko, Langevin and JW. Theorem Let R = Z/p k Z, p prime. Then R has EP for the Lee and the Euclidean weights. Foundational Aspects June 22, / 37

33 Sketch of proof (a) R = Z/p k Z is a chain ring. The symmetry group U = {±1}. det W factors into a product of character sums over G p k = U(R)/U. General character sums reduce to understanding Fourier transforms of w over G p j for (perhaps) smaller j k. The characters involved are primitive even characters modulo p j. Foundational Aspects June 22, / 37

34 Sketch of proof (b) Easy: ŵ L (χ) 0 and ŵ E (χ) 0 for χ = 1: sums of positive terms. The specific form of w L and w E, together with a Fourier transform calculation, implies that ŵ L (χ) = 0 if and only if ŵ E (χ) = 0 for every primitive even character χ 1. Foundational Aspects June 22, / 37

35 Sketch of proof (c) If ŵ L (χ) = 0 or ŵ E (χ) = 0, then the other also vanishes. Calculate that the second generalized Bernoulli number B 2 (χ) = 0. This implies that the Dirichlet L-function L(s, χ) vanishes at s = 1 for a primitive even character, which contradicts known behavior of L(s, χ). Thus ŵ L (χ) 0 and ŵ E (χ) 0, so det W 0 in both cases. Foundational Aspects June 22, / 37

36 Relation between the determinants For Z/2 k Z, det W E = 2 (k 1)2k 1 det W L. For odd primes there is a similar relation. Special case: when 2 is a primitive root modulo p k, k (2 pj 1 (p 1)/2 + 1) det W E = p k(pk 1)/2 det W L. j=1 Foundational Aspects June 22, / 37

37 What about Z/NZ? We expect w L and w E to have EP for all N. What s missing is a factorization of det W, or some other decomposition, for general N. Foundational Aspects June 22, / 37