The MacWilliams Identities

Transcription

1 The MacWilliams Identities Jay A. Wood Western Michigan University Colloquium March 1, 2012

2 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) MacWilliams Identities March 1, / 35

3 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) MacWilliams Identities March 1, / 35

4 Example: Hamming Code Alphabet is F 2 = Z/2Z. Let E 7 F 7 2 be spanned by the rows of Messages M form a vector space of dimension 4 over F 2, and E 7 corrects one error. JW (WMU) MacWilliams Identities March 1, / 35

5 Hamming Code, Continued x E 7 if and only if Hx T = 0, where H = If there is one error in x, then Hx T 0 and Hx T gives the binary representation of the position where the error occurs. H is the generator matrix for the dual code E7. JW (WMU) MacWilliams Identities March 1, / 35

6 Objectives for this Talk Some of the language of algebraic coding theory. Basic properties of dual codes, including the MacWilliams identities. The MacWilliams identities and their proof. Generalizations over finite rings. JW (WMU) MacWilliams Identities March 1, / 35

7 Definitions (a) Let F q be a finite field, q a prime power. For example, F p = Z/pZ, the integers modulo a prime p. The case p = 2 is especially important. A linear code of length n is a vector subspace C F n q. Can generalize to left submodules C R n, for a finite ring R with 1. Or even C A n, for a finite left module A over R. JW (WMU) MacWilliams Identities March 1, / 35

8 Definitions (b) For x = (x 1,..., x n ) F n q, the Hamming weight wt(x) equals the number of nonzero entries of x. For a linear code C F n q, the Hamming weight enumerator is the polynomial W C (X, Y ) = x C X n wt(x) Y wt(x). An estimate of the error-correcting capability of the code C is given by the minimum weight d C = min{wt(x) : x C, x 0}. (d E7 = 3.) JW (WMU) MacWilliams Identities March 1, / 35

9 Definitions (c) Define a dot product on F n q by x y = n x i y i, i=1 for x = (x 1,..., x n ), y = (y 1,..., y n ) F n q. For a linear code C F n, define the dual code C (also denoted r(c), the right annihilator) by C = {y F n q : x y = 0, x C}. JW (WMU) MacWilliams Identities March 1, / 35

10 Standard Properties of Dual Codes 1. C F n q. 2. C is a linear code. 3. dim C + dim C = n; or C C = F n q. 4. (C ) = C. 5. The MacWilliams identities hold: W C (X, Y ) = 1 C W C(X + (q 1)Y, X Y ). JW (WMU) MacWilliams Identities March 1, / 35

11 Proofs The first four items are obvious or follow from basic linear algebra of nondegenerate forms over fields. The proof of the MacWilliams identities given will be based on character theory and the Poisson summation formula. The proof is due to Gleason. JW (WMU) MacWilliams Identities March 1, / 35

12 Florence Jessie MacWilliams Worked for many years at Bell Labs doctoral dissertation under Andrew Gleason at Harvard. (Mackey, Stone, Ge. Birkhoff, EH Moore.) Combinatorial Problems of Elementary Abelian Groups The second section is the MacWilliams identities. JW (WMU) MacWilliams Identities March 1, / 35

13 An Eye Towards Generalizations When F q is replaced by a finite ring R, do the standard properties, including the MacWilliams identities, still hold? For the double dual, need R to be quasi-frobenius. For the size condition and the MacWilliams identities, need R to be Frobenius. Why Frobenius? There is a character-theoretic proof over F that uses the crucial property F = F. Frobenius rings satisfy R = R, and the same proof will work. JW (WMU) MacWilliams Identities March 1, / 35

14 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. If M is a finite left R-module, then M is a right R-module. (Thus, F = F, as vector spaces.) JW (WMU) MacWilliams Identities March 1, / 35

15 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) MacWilliams Identities March 1, / 35

16 Frobenius Rings 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) MacWilliams Identities March 1, / 35

17 Two Useful Theorems About Finite Frobenius Rings Finite ring R with 1. (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 (Hirano, 1997; indep. 1999). Corollary: R is Frobenius iff there exists a character ρ of R such that ker ρ contains no nonzero left (right) ideal of R. This ρ is a generating character. JW (WMU) MacWilliams Identities March 1, / 35

18 Examples of Finite Frobenius Rings Finite fields F q : χ(x) = exp(2πi Tr q,p (x)/p). Z/nZ: χ(x) = exp(2πix/n). Galois rings (Galois extensions of Z/p m Z). Soc(R) = F p m; use χ for F p m and extend. Finite chain rings (all ideals form a chain). Extend from Soc(R) = F q. Products of Frobenius rings. Use product of the generating characters. Matrix rings over a Frobenius ring: M n (R). χ = χ R Tr. Finite group rings over a Frobenius ring: R[G]. χ( a g g) = χ R (a e ), e is identity element. JW (WMU) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

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) MacWilliams Identities March 1, / 35

26 Example, n = 2 Let C 2 F 2 2 be C 2 = {00, 11}. C 2 is self-dual; i.e., C 2 = C 2. W C2 (X, Y ) = X 2 + Y 2. Call this S. In a binary self-dual code, all elements have even weight. JW (WMU) MacWilliams Identities March 1, / 35

27 Example, n = 8 Let E 8 F 8 2 be spanned by the rows of E 8 is self-dual. Also, all elements x of E 8 satisfy wt(x) 0 mod 4. E 8 is doubly-even. W E8 (X, Y ) = X X 4 Y 4 + Y 8. Call this T. JW (WMU) MacWilliams Identities March 1, / 35

28 Extended Golay Code (1949) There is a famous binary code G 24 of length 24, dimension 12, which is self-dual, doubly-even, with minimum weight 8. Weight enumerator: W G24 (X, Y ) = X X 16 Y X 12 Y X 8 Y 16 + Y 24. JW (WMU) MacWilliams Identities March 1, / 35

29 Gleason s Theorem (1970) The weight enumerator of any binary self-dual code is a polynomial expression in the weight enumerators S and T of C 2 and E 8. W G24 (X, Y ) = T (2S 8 T S 4 T 2 S 12 ). The weight enumerator of any binary self-dual, doubly-even code is a polynomial expression in the weight enumerators of E 8 and G 24. So, n 0 mod 8. Greatly generalized by Nebe, Rains, and Sloane in JW (WMU) MacWilliams Identities March 1, / 35

30 Self-Dual Codes in the Non-Commutative Setting For a left linear code C R n, R Frobenius, the right linear code r(c) is the proper choice for the dual code to C. There is no guarantee that a right code will also be a left code. The same for C A n and (Ân : C) Ân. How to make sense of self-duality? Follow ideas of Nebe, Rains, and Sloane. JW (WMU) MacWilliams Identities March 1, / 35

31 Making Identifications (a) Left-right identifications: assume the ring R admits an anti-isomorphism. Anti-isomorphism ε : R R, additive isomorphism, with ε(rs) = ε(s)ε(r), for all r, s R. Involution: if ε 2 = 1. Given a left R-module M, define a right R-module ε(m) to be the same additive group, but with right scalar multiplication mr := ε(r)m, for m M, r R. JW (WMU) MacWilliams Identities March 1, / 35

32 Making Identifications (b) Characters: assume the alphabet A admits an isomorphism ψ : ε(a) Â of right R-modules. (If A = R, this happens iff R is Frobenius.) Dual code: for a left linear code C A n, define the dual code C = ε 1 ψ 1 (Ân : C). The dual code is now a left submodule of A n. JW (WMU) MacWilliams Identities March 1, / 35

33 An Example: Group Algebras G finite group. R = F q [G], the group algebra. Involution ε( a g g) = a g g 1. Use A = R, which is Frobenius. Example: G = Σ 3, symmetric group: σ 3 = e, τ 2 = e, στ = τσ 2. Use q = 2. C = R(e + τ)(e + σ + σ 2 ) + R(e + σ + τσ + τσ 2 ) is a self-dual code of length 1. JW (WMU) MacWilliams Identities March 1, / 35

34 Another Example: Subalgebras of the Steenrod Algebra The Steenrod algebra is an (infinite) Frobenius F p -algebra that arises in algebraic topology. The F p -cohomology of any CW-complex is a module over the Steenrod algebra. For finite examples, use some subhopf algebras. For example, when p = 2, use A(1), the subalgebra generated by 1, Sq 1, and Sq 2. Then A = H (RP 4k+3 ; F 2 ) admits an appropriate ψ. JW (WMU) MacWilliams Identities March 1, / 35

35 Questions Which finite rings admit anti-isomorphisms? Which finite modules admit isomorphisms ψ : ε(a) Â? Are there interesting examples of self-dual codes? There are some partial results (2010), but the area is wide open. JW (WMU) MacWilliams Identities March 1, / 35