Symmetries of Weight Enumerators

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1 Martino Borello (Paris 8-LAGA) Trento, / 23 Symmetries of Weight Enumerators Martino Borello Université Paris 8 - LAGA Trento,

2 Martino Borello (Paris 8-LAGA) Trento, / 23 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) Trento, / 23 Background Background q a prime power. Basic definitions 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 x, y : F n q F n q Ñ F q is the standard inner product, C K : tv P F n q xv, cy 0, for all c P Cu pdual of Cq. If C C K, the code C is called self-dual.

4 Background Martino Borello (Paris 8-LAGA) Trento, / 23 Weight enumerator C F n q ù w C px, yq : cpc with A i : #tc P C wtpcq iu. x n wtpcq y wtpcq ņ i0 A i x n i y i 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? Remark: self-dual ñ even., x y? 2 2.

5 Background Martino Borello (Paris 8-LAGA) Trento, / 23 Group action GL 2 pcq ý Crx, ys: a A : c b d, ppx, yq ÞÑ ppx, yq A : ppax by, cx dyq. For G GL 2 pcq, denote Crx, ys G : tppx, yq ppx, yq A ppx, P Gu. For ppx, yq P Crx, ys, denote Spppx, yqq : ta P GL 2 pcq ppx, yq A ppx, yqu GL 2 pcq. Example G : B F 1 0 ñ Crx, ys G Crx, y 2 s. 0 1

6 Martino Borello (Paris 8-LAGA) Trento, / 23 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.,

7 Gleason's Theorem Martino Borello (Paris 8-LAGA) Trento, / 23 C F n 2 self-dual and doubly-even. Consequences 8 n (Gleason '71). Y n d : min wtpcq 4 cpc t0u 24] If the bound is achieved C is called extremal. A length n is called jump length if 24 n. extremal and doubly-even ñ n (Mallows and Sloane '73). (Zhang '99). Other results loooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooon jump length and extremal ñ doubly-even (Rains '98) ó all codewords of given weight support a 5-design (Assmus and Mattson '69)

8 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, Idea: properties of families of (self-dual) codes ù symmetries of weight enumerators ù new properties. Our questions Given a weight enumerator of a code, which are its symmetries? Are they shared by the whole family of this code? Which are the possible groups of symmetries? Can we determine with these methods unknown weight enumerators? Martino Borello (Paris 8-LAGA) Trento, / 23

9 Martino Borello (Paris 8-LAGA) Trento, / 23 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. This is a set of N degpppx, yqq 1 points. π : Spppx, yqq GL 2 pcq ÞÑ Spppx, yqq PGL 2 pcq. induces a b, px : yq c d PGL 2 pcq ÞÑ Spppx, yqq ý P 1 pcq pax by : cx dyq V pppx, yqq. ý simply 3-transitive

10 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 ppx, yq P Crx, ys h of degree n. Remark 1 We have Then ppx, yq ppλx, λyq ô λ n 1. NB F ζn 0 Spppx, yqq Spppx, yqq, 0 ζ n where ζ n P C is a primitive n-th root of unity. If Spppx, yqq 8, then Remark 2 #Spppx, yqq n #Spppx, yqq. For all A P GL 2 pcq, we have Spppx, yq A q Spppx, yqq A.

11 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 Theorem (B.,Mila) #Spppx, yqq 8 ô #V pppx, yqq 3. Proof: ðq V pppx, yqq tp 1, P 2, P 3,..., P m j, ku t1,..., mu there is at most one element A P Spppx, yqq s.t. P A 1 P i, P A 2 P j, P A 3 P k. Then Spppx, yqq m pm 1q pm 2q. ñq If #V pppx, yqq 3, then #Spppx, yqq 8. l In particular, if #V pppx, yqq 3, then #Spppx, yqq n 4.

12 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 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, yq A q is a central extension of one of the groups listed above.

13 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 C F n q. Proposition ζ m P SpwC px, yqq ô wtpcq mz (divisibility). In particular, if m 5, wtpcq mz ñ Spw C px, yqq C m 1 or Spw C px, yqq D m 1 (m m 1 ). Lemma If q 2, r s P Spw Cpx, yqq ô 1 p1, 1,..., 1q P C. Example (Repetition code) C the r12, 2, 6s binary code with generator matrix r s w C px, yq x 12 2x 6 y 6 y 12 ù Spw C px, yqq D 6.

14 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 MacWilliams identities. Example (ternary Golay code) C the r12, 6, 6s 3 ternary code with generator matrix w C px, yq x x 6 y 6 440x 3 y 9 24y 12 ù Spw C px, yqq A 4. Example (Hamming code) C the r8, 4, 4s binary code with generator matrix w C px, yq x 8 14x 4 y 4 y 8 ù Spw C px, yqq S 4.

15 Possible symmetries Martino Borello (Paris 8-LAGA) Trento, / 23 Example f 1 px, yq : x x 15 y 5 494x 10 y x 5 y 15 y 20 ; f 2 px, yq : x x 25 y x 20 y x 10 y x 5 y 25 y 30. ppx, yq P Crf 1 px, yq, f 2 px, yqs ñ Spppx, yqq A 5. Open problem Is there a code C such that Spw C px, yqq A 5? Extensive search in Crf 1 px, yq A, f 2 px, yq A s for A P GL 2 pcq of ppx, yq s.t. its coecients are positive; pp1, 0q 1; pp1, 1q is a prime power. Not yet found.

16 Martino Borello (Paris 8-LAGA) Trento, / 23 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.

17 Martino Borello (Paris 8-LAGA) Trento, / 23 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.

18 The algorithm Martino Borello (Paris 8-LAGA) Trento, / 23 C F n q linear code. Remark 1 0 P C ñ w C px, yq x n... ñ p1 : 0q R V pw C px, yqq. Remark 2 w C px, yq P Zrx, ys ñ the roots of w C px, 1q are in Z. In the algorithm, roots in K s.t. rk : Qs 8 (splitting eld). Remark 3 If we consider roots in C, we have to deal with approximations.

19 Martino Borello (Paris 8-LAGA) Trento, / 23 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

20 Reed-Muller codes Martino Borello (Paris 8-LAGA) Trento, / 23 Theorem (Ax '64) Let : qt m 1 r u. Then wtprm q pr, mqq Z. Lemma If r mpq 1q, then RM q pr, mq K RM q pmpq 1q r 1, mq. Remark RM q pr, mq self-dual ô $ & % q power of 2, m is odd, r mpq 1q 1 2. In particular, RM 2 pr, 2r 1q self-dual and doubly-even ñ Spw RM2pr,2r 1qpx, yqq S 4.

21 Reed-Muller codes Martino Borello (Paris 8-LAGA) Trento, / 23 Ax's theorem implies: 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 is either cyclic or dihedral. Proof ζ P SpwRMqpr,mqpx, yqq of order 5 (ζ primitive -th root of unity). l

22 Reed-Muller codes By the algorithm we get: Theorem (B.,Mila) If m 2, then u u 1 u 1 u P SpwRM2pm 1,mqpx, yqq, with u : ζ 1 2 (ζ primitive 2 m -th root of unity). Theorem (B.,Mila) Let C P trm 4 p2, 2q, RM 4 p3, 2q, RM 5 p2, 2qu, then Spw C px, yqq tidu. Open problem Understand the general behavior and deduce properties and new weight enumerators. Martino Borello (Paris 8-LAGA) Trento, / 23

23 At most two roots 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) Trento, / 23

24 At most two roots Martino Borello (Paris 8-LAGA) Trento, / 23 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 :

25 At most two roots Martino Borello (Paris 8-LAGA) Trento, / 23 Thank you very much for the attention!

Symmetries of weight enumerators

Martino Borello (Paris 8-LAGA) Gaeta, 06.06.2017 0 / 13 Symmetries of weight enumerators Martino Borello Université Paris 8 - LAGA Fq13 Martino Borello (Paris 8-LAGA) Gaeta, 06.06.2017 1 / 13 Introduction