On Extremal Codes With Automorphisms

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1 On Extremal Codes With Automorphisms Anton Malevich Magdeburg, 20 April 2010 joint work with S. Bouyuklieva and W. Willems 1/ 33

2 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes 4. Automorphisms of extremal codes 2/ 33

3 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes 4. Automorphisms of extremal codes 3/ 33

4 Introduction Linear code C is a k-dim subspace of an n-dim vector space Generator matrix G consists of basis vectors of C a 11 a 12 a 1n G = a 21 a 22 a 2n a k1 a k2 a kn Parity check matrix H (size n (n k)): Hx = 0 for all x C 4/ 33

5 Introduction Weight of a codeword is the number of its nonzero coordinates Weight enumerator: W C (x, y) = u C x n wt(u) y wt(u) = n A i x n i y i i=0 Parameters [n, k, d] stand for length, dimension and minimum distance. 5/ 33

6 Automorphism Group Definition Aut(C) = {σ S n uσ C for all u C} G = Aut(C) If G is trivial, C is only a vector space If G is nontrivial, it may help to construct the code. C is a module for G 6/ 33

7 Types of Automorphisms Definition σ Aut(C) is called of type p-(c, f ) if it has exactly c cycles of length p and f fixed points. If n is the length of C then pc + f = n. c {}}{ ( } {{ } ) ( } {{ } )... ( } {{ } ) ( )... ( ) }{{} p p p f 7/ 33

8 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes 4. Automorphisms of extremal codes 8/ 33

9 The dual code C = {u u v = 0 for all v C} The dual weight enumerator W C (x, y) = 1 C W C(x + y, x y) If C = C the code is self-dual For a self-dual code k = n/2 and all codewords have even weight 9/ 33

10 Self-Dual Codes Two types of self-dual codes: Type I (SE): all weights are even Type II (DE): all weights are a multiple of 4 Theorem (Gleason) Weight enumerator of a self-dual code is a polynomial in f and g of degrees respectively for Type I codes: 2 and 8, for Type II codes: 8 and / 33

11 Self-Dual Codes Two types of self-dual codes: Type I (SE): all weights are even Type II (DE): all weights are a multiple of 4 Corollary Length of a Type II code is a multiple of 8 n = 24m + 8i, i = 0, 1 or 2 10/ 33

12 Extremality Corollary (Mallows, Sloane) d 4 n Type II (tight) d 2 n Type I (NOT tight) Theorem (Rains, using shadow) New bounds for Type I codes are: d 4 n n 22 (mod 24) d 4 n n 22 (mod 24) 11/ 33

13 Zhang: no extremal DE codes for n > 3952 Lengths of known extremal codes: 8, 16, 24, 32, 40, 48, 56, 64, 80, 88, 104, 112, 136 For red lengths extended quadratic residue codes are extremal. For blue lengths quadratic double circulant codes are extremal. 12/ 33

14 Existing Codes [24, 12, 8] Golay code Aut(C) = M 24 (5-transitive) Order of Aut(C): Unique [48, 24, 12] QR-code Aut(C) = PSL 2 (47) (2-transitive) Order of Aut(C): Unique (computer search) 13/ 33

15 Putative Codes [72, 36, 16]-code Aut(C) is solvable Possible primes in Aut(C) : 2, 3, 5, 7 Possible order: Aut(C) 36 [96, 48, 20]-code Possible primes in Aut(C) : 2, 3, 5 14/ 33

16 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes 4. Automorphisms of extremal codes 15/ 33

17 Cyclic Codes a 0 a 1 a p G = a p a 0 a p a 1 a 2 a 0 C is an ideal in F 2 [x]/ x p 1 C = e(x) with e(x) = e 2 (x) Automorphisms of C: cyclic shift σ of order p µ: x x 2 of order s(p) 16/ 33

18 Definition of QR Codes p ±1 mod 8 Quadratic residues Q and nonresidues N QR codes are cyclic codes with e(x) = n N x n x r or e(x) = r Q 17/ 33

19 Extended QR Codes If p 1 mod 8: Extended QR code C is self-dual of Type II Aut(C) = PSL 2 (p), p 7 or 23 Aut(C) is 2-transitive Aut(C) = p (p 1)(p+1) 2 18/ 33

20 Importance of QR Codes Hamming and Golay codes are QR Large minimum distance: d 2 d + 1 p (if p ±1 mod 8) Asymptotically good? Different decoding techniques 19/ 33

21 Quadratic Double Circulant Codes n = 2q + 2, q 3 mod 8 is a prime G = Q 1 1 Q cyclic matrix, corr. to residues C is self-dual of Type II Aut(C) = PSL 2 (q) Z 2 20/ 33

22 1. Linear codes 2. Self-dual and extremal codes 3. Quadratic residue codes 4. Automorphisms of extremal codes 21/ 33

23 Motivation n d p Aut(C) comment 8 4 2, 3, 7 xqr, QDC , 3, 5, 7, 11, 23 xqr, QDC , 3, 5, 7, 31 xqr , 3, 5, 7, 19 QDC , 3, 23, 47 xqr , 5, 19, 79 xqr , 3, 7, 11, 43 QDC , 3, 13, 17, 103 xqr , 7 Harada, , 3, 11, 67 QDC 22/ 33

24 Possible Automorphisms Theorem C is an extremal self-dual code C is of length n 48 σ Aut(C) is of type p (c, f ) where p 5 is a prime. Then c f. 23/ 33

25 Assumptions C is self-dual, doubly-even, extremal σ Aut(C) of prime order p > n/2 Corollary c = f = 1 since n = pc + f σ is of type p (1, 1) n = p + 1 = 24m + 8i, i = 0, 1 i 2 since 24m is not a prime 24/ 33

26 Definition s(p) denotes the smallest number s N, such that p 2 s 1. In our case: s(p) = p 1, k 2 even k Proposition If k = 2 then C is an extended QR code. 25/ 33

27 Definition s(p) denotes the smallest number s N, such that p 2 s 1. In our case: s(p) = p 1, k 2 even k Proposition If k = 2 then C is an extended QR code. 25/ 33

28 Classification of QR Codes Theorem Let C be a self-dual doubly-even extended QR code. Then C is extremal exactly for the lengths 8, 24, 32, 48, 80 and / 33

29 Sketch of the proof Task: find a codeword of weight < 4 n in a large code C. Way out: search in a suitable subcode C < C. H < Aut(C) C = C H = {codewords fixed by H} 27/ 33

30 Sketch of the proof C extended QR of length n = p + 1 G = Aut(C) = PSL(2, p) H < G H = cyclic of order 4 or 6 H = Syl 2 (G) 28/ 33

31 Cases depending on k n = 24m or n = 24m + 8 p = n 1 s(p) = p 1 k k = 2 (one code) solved ( ) k > 2 codes open 2 k/2 k similar method may be applied 29/ 33

32 Case k > 2 Possibly large number of codes C is an extended cyclic code automorphisms of order p and s(p) Aut(C) is not transitive! if s(p) prime difficulties 30/ 33

33 List of Open Cases p s(p) k codes d / 33

34 Conjecture There are no extremal self-dual doubly-even codes having an automorphism of prime order p > n/2 apart from the cases n = 8, 24, 32, 48, 80 and / 33

35 Summary C is self-dual, doubly-even, extremal σ Aut(C) of prime order p > n/2 We give full classification of extremal extended QR codes that solves the case s(p) = p 1 2 For s(p) < p 1 2 some cases still remain open 33/ 33

New binary self-dual codes of lengths 50 to 60

Designs, Codes and Cryptography manuscript No. (will be inserted by the editor) New binary self-dual codes of lengths 50 to 60 Nikolay Yankov Moon Ho Lee Received: date / Accepted: date Abstract Using