Applications of Finite Frobenius Rings to Algebraic Coding Theory I. Two Theorems of MacWilliams over Finite Frobenius Rings

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1 Applications of Finite Frobenius Rings to Algebraic Coding Theory I. Two Theorems of MacWilliams over Finite Frobenius Rings Jay A. Wood Western Michigan University jwood Symposium on Ring and Representation Theory, Okayama University, September 25, 2011

2 Acknowledgments I am pleased to be visiting the city of Okayama, and I thank the symposium organizers, especially Professor Kunio Yamagata, for the invitation and for their hospitality. I thank the Japan Society for the Promotion of Science for its financial support of the symposium. JW (WMU) Theorems of MacWilliams September 25, / 30

3 For Your Amusement Table: Mathematical Genealogy of Two Speakers Dan Zacharia Jay Wood Maurice Auslander Shoshichi Kobayashi Robert Taylor Carl Allendoerfer JHC Whitehead Tracy Thomas Oswald Veblen EH Moore. Simeon Poisson Pierre-Simon Laplace (and JL Lagrange) Jean Le Rond d Alembert JW (WMU) Theorems of MacWilliams September 25, / 30

4 The Coding Problem How to ensure the integrity of a message transmitted over a noisy channel? Cleverly add redundancy. Encode possible messages (information) as a string of elements in an alphabet. Transmit the string over the channel. Detect errors and decode. JW (WMU) Theorems of MacWilliams September 25, / 30

5 Adding Algebraic Structure Assume the alphabet is a finite field F. Assume the set of messages M is a finite dimensional vector space over F of dimension k. The encoding is a linear embedding M F n, for some n. The image is a linear code of length n. JW (WMU) Theorems of MacWilliams September 25, / 30

6 Objectives for this Talk Some of the language of algebraic coding theory. Two theorems of MacWilliams valid over finite fields. Finite Frobenius rings and their character modules. Generalize the two theorems. JW (WMU) Theorems of MacWilliams September 25, / 30

7 Definitions (a) Let R be a finite associative ring with 1. Let A be a finite unital left R-module; A will be the alphabet. A left linear code over A of length n is a left R-submodule C A n. Special case: when A = R as a left module. JW (WMU) Theorems of MacWilliams September 25, / 30

8 Definitions (b) For x = (x 1,..., x n ) A n, the Hamming weight wt(x) equals the number of nonzero entries of x. For a linear code C A n, the Hamming weight enumerator is the polynomial W C (X, Y ) = x C X n wt(x) Y wt(x). JW (WMU) Theorems of MacWilliams September 25, / 30

9 Definitions (c) When A = R, define a dot product on R n by x y = n x i y i, i=1 for x = (x 1,..., x n ), y = (y 1,..., y n ) R n. When A = R, and C R n is a left linear code, define the right annihilator r(c) by r(c) = {y R n : x y = 0, x C}. JW (WMU) Theorems of MacWilliams September 25, / 30

10 MacWilliams Identities (1962/63) Let C F n q be a linear code over F q. The MacWilliams identities hold: W C (X, Y ) = 1 r(c) W r(c)(x + (q 1)Y, X Y ). JW (WMU) Theorems of MacWilliams September 25, / 30

11 Florence Jessie MacWilliams doctoral dissertation under Andrew Gleason at Harvard. (Mackey, Stone, Ge. Birkhoff, EH Moore.) Combinatorial Problems of Elementary Abelian Groups Three sections: Extension theorem on isometries The MacWilliams identities Coverings JW (WMU) Theorems of MacWilliams September 25, / 30

12 Monomial Transformations A monomial transformation T : R n R n has the form T (x 1,..., x n ) = (x σ(1) u 1,..., x σ(n) u n ), where σ is a permutation of {1,..., n} and u 1,..., u n are units of R. A monomial transformation preserves Hamming weight: wt(t (x)) = wt(x), x R n. JW (WMU) Theorems of MacWilliams September 25, / 30

13 MacWilliams Extension Theorem (1961/62) Assume R = F, a finite field. Assume C 1, C 2 F n are linear codes. If f : C 1 C 2 is a linear isomorphism that preserves Hamming weight, then f extends to a monomial transformation of F n. JW (WMU) Theorems of MacWilliams September 25, / 30

14 Generalizing the Theorems of MacWilliams When A = R, are the MacWilliams identities and the MacWilliams extension theorem still valid? Yes, if R is a finite Frobenius ring. Why Frobenius? There are character-theoretic proofs over finite fields that use the crucial property F = F. Frobenius rings satisfy R = R, and the same proofs will work. JW (WMU) Theorems of MacWilliams September 25, / 30

15 Characters Let (G, +) be a finite abelian group. A character π of G is a group homomorphism π : (G, +) (C, ), the nonzero complexes. The set Ĝ of all characters of G is itself a finite abelian group called the character group. Ĝ = G. As elements of the vector space of all functions from G to C, the characters are linearly independent. If M is a finite left R-module, then M is a right R-module. JW (WMU) Theorems of MacWilliams September 25, / 30

16 Two Useful Formulas π(x) = x G π(x) = π Ĝ { G, π = 1, 0, π 1. { G, x = 0, 0, x 0. JW (WMU) Theorems of MacWilliams September 25, / 30

17 Finite Frobenius Rings Finite ring R with 1. The (Jacobson) radical Rad(R) of R is the intersection of all maximal left ideals of R; Rad(R) is a two-sided ideal of R. The (left/right) socle Soc(R) of R is the ideal of R generated by all the simple left/right ideals of R. R is Frobenius if R/ Rad(R) = Soc(R) as one-sided modules (both left and right). JW (WMU) Theorems of MacWilliams September 25, / 30

18 Two Useful Theorems About Finite Frobenius Rings (Honold, 2001) R/ Rad(R) = Soc( R R) as left modules iff R/ Rad(R) = Soc(R R ) as right modules. R is Frobenius iff R = R as left modules iff R = R as right modules (1999). Corollary: R is Frobenius iff there exists a character π of R such that ker π contains no nonzero left (right) ideal of R. This π is a generating character. JW (WMU) Theorems of MacWilliams September 25, / 30

19 Fourier Transform Given a function f : G V, with V a complex vector space, its Fourier transform is a function ˆf : Ĝ V defined by ˆf (π) = x G π(x)f (x), π Ĝ. Fourier inversion: f (x) = 1 G π( x)ˆf (π), x G. π Ĝ JW (WMU) Theorems of MacWilliams September 25, / 30

20 Poisson Summation Formula For a subgroup H G, define its annihilator (Ĝ : H) = {π Ĝ : π(h) = 1}. (Ĝ : H) = G / H. For a subgroup H G and any a G, 1 f (a + h) = h H (Ĝ : H) π( a)ˆf (π). π (Ĝ:H) In particular, for a subgroup H G, 1 f (h) = ˆf h H (Ĝ : H) (π). π (Ĝ:H) JW (WMU) Theorems of MacWilliams September 25, / 30

21 MacWilliams Identities over Finite Frobenius Rings Theorem (1999) Let R be a finite Frobenius ring. If C R n is a left linear code, then the MacWilliams identities hold: W C (X, Y ) = 1 r(c) W r(c)(x + ( R 1)Y, X Y ). JW (WMU) Theorems of MacWilliams September 25, / 30

22 Proof of the MacWilliams Identities (a) The proof follows a proof due to Gleason (1970). Let R be Frobenius with generating character ρ. Let G = R n, an abelian group under addition. Let H = C, a left linear code. Let V = C[X, Y ], a complex vector space. Let f : G V be f (x) = X n wt(x) Y wt(x). JW (WMU) Theorems of MacWilliams September 25, / 30

23 Proof of the MacWilliams Identities (b) By Frobenius hypothesis, every character of G = R n has the form π a, for some a R n, with π a (x) = ρ(x a), x R n. π a (Ĝ : H) if and only if a r(c). (Ĝ : H) = r(c). JW (WMU) Theorems of MacWilliams September 25, / 30

24 Proof of the MacWilliams Identities (c) For f (x) = X n wt(x) Y wt(x), ˆf (π a ) = (X + ( R 1)Y ) n wt(a) (X Y ) wt(a). This requires some manipulations and use of π(x) formulas. (Next slide.) Recognize ˆf (π a ) as summand of W r(c) (X + ( R 1)Y, X Y ). JW (WMU) Theorems of MacWilliams September 25, / 30

25 Idea of Manipulation Let n = 1, f (x) = X 1 wt(x) Y wt(x). ˆf (π a ) = x R π a (x)x 1 wt(x) Y wt(x) = X + x 0 π a (x)y = { X + ( R 1)Y, a = 0, X Y, a 0, = (X + ( R 1)Y ) 1 wt(a) (X Y ) wt(a) JW (WMU) Theorems of MacWilliams September 25, / 30

26 MacWilliams Extension Theorem over Finite Frobenius Rings Theorem (1999) Let R be a finite Frobenius ring, and suppose C 1, C 2 R n are left linear codes. If f : C 1 C 2 is an R-linear isomorphism that preserves Hamming weight, then f extends to a monomial transformation of R n. JW (WMU) Theorems of MacWilliams September 25, / 30

27 Character-Theoretic Proof (a) The proof follows a proof of Ward and Wood in the finite field case (1996). View C i as the image of λ i : M R n, with λ i = (λ i,1,..., λ i,n ) and λ 2 = f λ 1. Using character sums, express Hamming weight as: wt(λ i (x)) = n n j=1 1 R π(λ i,j (x)), x M. π R JW (WMU) Theorems of MacWilliams September 25, / 30

28 Character-Theoretic Proof (b) Because f preserves Hamming weight, we get n π(λ 1,j (x)) = j=1 π R n ψ(λ 2,k (x)), x M. k=1 ψ R In a Frobenius ring, there is a generating character ρ. Every character of R has the form aρ, a R. (aρ)(r) := ρ(ra), r R. JW (WMU) Theorems of MacWilliams September 25, / 30

29 Character-Theoretic Proof (c) Re-write weight-preservation equation as n (aρ)(λ 1,j (x)) = j=1 a R n (bρ)(λ 2,k (x)), x M. k=1 b R Or as n ρ(λ 1,j (x)a) = j=1 a R n ρ(λ 2,k (x)b), x M. k=1 b R JW (WMU) Theorems of MacWilliams September 25, / 30

30 Character-Theoretic Proof (d) The last equation is an equation of characters on M. Characters are linearly independent, so one can match up terms (carefully). A technical argument involving a preordering given by divisibility in R shows how to match up terms with units as multipliers. This produces a permutation σ and units u i in R such that λ 2,k = λ 1,σ(k) u k, as desired. JW (WMU) Theorems of MacWilliams September 25, / 30