Symmetries of weight enumerators

Transcription

1 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Symmetries of weight enumerators Martino Borello Université Paris 8 - LAGA Fq13

2 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Introduction Introduction One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. N. Sloane Properties of codes (or of families of codes) c c ú Symmetries of their weight enumerators M. Borello, O. Mila. On the Stabilizer of Weight Enumerators of Linear Codes. arxiv:

3 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Background Background q a prime power. A q-ary linear code C of length n is a subspace of F n q. If c pc 1,..., c n q P C (codeword), the (Hamming) weight of c is wtpcq : #ti P t1,..., nu c i 0u (wtpcq : twtpcq c P Cu). If C C K, the code C is called self-dual. C F n q ù w C px, yq : cpc x n wtpcq y wtpcq ņ i0 A i x n i y i with A i : #tc P C wtpcq iu (weight enumerator of C).

4 Background Martino Borello (Paris 8-LAGA) Gaeta, / 13 C binary linear code. Divisibility conditions Even: wtpcq 2Z ñ w C px, yq w C px, yq. Doubly-even: wtpcq 4Z ñ w C px, yq w C px, iyq. MacWilliams' identities Self-dual ñ w C px, yq w x y C? Group action GL 2 pcq ý Crx, ys: ppx, yq a b c d, x y? 2 2. For G GL 2 pcq, the invariant ring of G is : ppax by, cx dyq. Crx, ys G : tppx, yq ppx, yq A ppx, P Gu. Notation: for ppx, yq P Crx, ys, Spppx, yqq : Stab GL2pCqpppx, yqq.

5 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Gleason's Theorem Gleason's Theorem Theorem (Gleason '70) Let C be a binary linear code which is self-dual and doubly-even. Then w C px, yq P Crf 1, f 2 s where f 1 : wĥ3 px, yq and f 2 : w G24 px, yq. C self-dual ñ w C px, yq w x y C?, x y? 2 2 C doubly-even ñ w C px, yq w C px, iyq, B F G :?, ñ Crx, ys i G Crf 1, f 2 s.,

6 Gleason's Theorem Martino Borello (Paris 8-LAGA) Gaeta, / 13 C F n 2 self-dual and doubly-even. Consequences 8 n (Gleason '71). dpcq 4 X n 24 \ 4 (Mallows and Sloane '73). If the bound is achieved C is called extremal. extremal and doubly-even ñ n 3928 (Zhang '99). Is there an extremal self-dual code of length 72? 24 looooooooooomooooooooooon n and extremal (Sloane '73). ó all codewords of given weight support a 5-design (Assmus and Mattson '69)

7 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Questions Questions Many generalization of Gleason's theorem. G. Nebe, E.M. Rains, N.J.A. Sloane. Self-dual codes and invariant theory. Vol. 17. Berlin: Springer, What if MacWilliams' identities do not give a symmetry? Our questions Which are the possible groups of symmetries? Given a weight enumerator of a code, which are its symmetries? Are they shared by the whole family of this code? Can we determine with these methods unknown weight enumerators?

8 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Possible symmetries Possible symmetries For ppx, yq P Crx, ys h (h=homogeneous), denote V pppx, yqq : tpx : yq P P 1 pcq ppx, yq 0u. π : Spppx, yqq GL 2 pcq ÞÑ Spppx, yqq PGL 2 pcq. induces a b, px : yq c d Theorem (B.,Mila) PGL 2 pcq ÞÑ Spppx, yqq ý P 1 pcq pax by : cx dyq V pppx, yqq. ý simply 3-transitive #Spppx, yqq 8 ô #V pppx, yqq 3.

9 Possible symmetries Martino Borello (Paris 8-LAGA) Gaeta, / 13 Theorem (Blichfeldt 1917) If H PGL 2 pcq is nite, then H is conjugate to one of the ζ m D Cm for a certain m P , i i D 1 1 A4. 0 ζ m, r 0 1 sd D 1 0 m for a certain m P r i s, i Corollary D 1 1 S4., i i 1 1, 2 ω ω 2 D A5 where ω p1? 5qi p1? 5q. If #V pppx, yqq 3, then DA P GL 2 pcq s.t. Spppx, yqq A is a central extension of one of the groups listed above.

10 Martino Borello (Paris 8-LAGA) Gaeta, / 13 The algorithm The algorithm Input: ppx, yq P Crx, ys h of degree n s.t. pp1, 0q G : H. 2. V :RootsOfpppx, 1qq tx 1,..., x m u. 3. If m 3, then print(innite group) and break; else V 3 : tall ordered 3-subsets of V u. 4. For tx 1, x 1, x 1 u P V : $ & x 1a b x1x 1 1c x1d 1 0 4a. Solve x 2a b x2x 1 2c x2d 1 0 (the unknowns are a, b, c, d). % x 3a b x3x 1 3c x3d 1 0 Call! a, b, c, d one ) of the 8 1 solutions. (simply 3-transitive) ax b 4b. If cx d x P V V, then a b 4bi. A : c d. 4bii. λ : ppa,cq pp1,0q. B : λ 1{n A. (to x the polynomial, not only the roots) 4biii. If ppx, y q B ppx, y q, then G : G Y tζ nb ζ n P C s.t. ζ n n 1u. Output: G Spppx, yqq.

11 Martino Borello (Paris 8-LAGA) Gaeta, / 13 The algorithm The algorithm Input: ppx, yq P Crx, ys h of degree n s.t. pp1, 0q G : H. 2. V :RootsOfpppx, 1qq tx 1,..., x m u. (Where?) 3. If m 3, then print(innite group) and break; else V 3 : tall ordered 3-subsets of V u. (#V 3 m 3 3m 2 2m) 4. For tx 1, x 1, x 1 u P V : $ & x 1a b x1x 1 1c x1d 1 0 4a. Solve x 2a b x2x 1 2c x2d 1 0 (the unknowns are a, b, c, d). % x 3a b x3x 1 3c x3d 1 0 Call! a, b, c, d one ) of the 8 1 solutions. (simply 3-transitive) ax b 4b. If cx d x P V V, then a b 4bi. A : c d. 4bii. λ : ppa,cq pp1,0q. B : λ 1{n A. (to x the polynomial, not only the roots) 4biii. If ppx, y q B ppx, y q, then G : G Y tζ nb ζ n P C s.t. ζ n n 1u. Output: G Spppx, yqq.

12 Martino Borello (Paris 8-LAGA) Gaeta, / 13 Reed-Muller codes Reed-Muller codes RM q pr, mq : tpf paqq apf m q Dimension and minimum distance known. Weight enumerator of a RM q pr, mq code f P F q rx 1,..., x m s of degree ru F qn q. c c ú Counting F q-rational points of hypersurfaces in A m pf q q N. Kaplan. Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory Thesis (Ph.D.) - Harvard University

13 Reed-Muller codes Martino Borello (Paris 8-LAGA) Gaeta, / 13 Remark RM 2 pr, 2r 1q self-dual and doubly-even ñ Spw RM2pr,2r 1qpx, yqq S 4. Theorem (B.,Mila) If one of the following holds q 2 and m 3r 1, q P t3, 4, 5u and m 2r 1, q 5 and m r 1, then Spw RMqpr,mqpx, yqq and Spw RMqpmpq 1q r 1,mqpx, yqq are cyclic or dihedral. Theorem (B.,Mila) Spw RM4p1,1qpx, yqq A 3? ? 15? E 1 3, 1 1 V 4 so that w RM4p1,1qpx, yq P Crf 1, f 2 s, where f 1 : 2x 2?? p3 15qxy p3 15qy 2, f 2 : 53x 4 36x 3 y 18x 2 y 2 636xy 3 213y 4.

14 Reed-Muller codes Martino Borello (Paris 8-LAGA) Gaeta, / 13 Table: Spw RM2 pr,mqpx, yqq m r D 4 D 8 D 16 D 32 D 64 D D 4 S 4 D 8 D 16 D 32 D D 8 D 8 S 4 D 4 D D 16 D 16 D 4 S D 32 D 32 D D 64 D D 128

15 Reed-Muller codes Martino Borello (Paris 8-LAGA) Gaeta, / 13 Table: Spw RM3 pr,mqpx, yqq m r D 3 D 9 D 27 D 81 1 D 3 C 3 C 9 C C 3 C 3 C D 9 C 3? C 9? D 27 C C D 81 Table: Spw RM4 pr,mqpx, yqq m r D 8 D 16 D 64 1 V 4 C 4 C 16 2 D 8 tidu C tidu tidu 4 8 C 4? 5 8 D 16 tidu C C D 64 Open problem Understand the general behavior and deduce properties and new weight enumerators.

16 Thanks Martino Borello (Paris 8-LAGA) Gaeta, / 13 Thank you very much for the attention!

17 Appendix A At most two roots C F n q s.t. #V pw Cpx, yqq 3. Theorem (B.,Mila) One of the following holds: C t0u; C F n q ; n is even and C is equivalent to n{2 à i1 r1, 1s; n is even, q 2 and w C px, yq px 2 y 2 q n{2. Open problem Is it possible to classify all the binary codes of even length n with weight enumerator px 2 y 2 q n{2? Martino Borello (Paris 8-LAGA) Gaeta, / 13

18 Appendix A Martino Borello (Paris 8-LAGA) Gaeta, / 13 M : tbinary codes of length n and weight enumerator px 2 y 2 q n{2 n P 2Nu{, Lemma pm, `q is a semigroup. the r2, 1, 2s code X 1 with generator matrix r1, 1s; the r6, 3, 2s code X 2 with generator matrix ; three r14, 7, 2s codes, X 3, X 4 and X 5, with generator matrices ri X 3 s,ri X 4 s and ri X 5 s respectively, where X 3 : , X 4 : and I is the 7 7 identity matrix Minimal set of generators? Innitely many?, X 5 :

19 Appendix B Martino Borello (Paris 8-LAGA) Gaeta, / 13 Another open problem Examples rn, 1, ns repetition code with n 3: w C px, yq x n y ù n Spw C px, yqq D n. r12, 6, 6s 3 ternary Golay code: w C px, yq x x 6 y 6 440x 3 y 9 24y ù 12 Spw C px, yqq A 4. r8, 4, 4s extended Hamming code: w C px, yq x 8 14x 4 y 4 y ù 8 Spw C px, yqq S 4. Open problem Is there a code C such that Spw C px, yqq A 5?