Some optimal self-orthogonal and self-dual codes

Abstract

The binary optimal self-orthogonal codes of dimension 4 k 10 and length 15 n k + 16 are classified. They are used for constructing self-dual codes of lengths 40, 42, and 44. All optimal binary self-dual codes of lengths 42 and 44 which have an automorphism of order 5 with four independent cycles are obtained up to equivalence. All optimal self-dual codes of lengths 40, 42 and 44 which have an automorphism of order 3 with six independent cycles are constructed. Some of them are the first known codes with their weight enumerators.

Self-dual codes with automorphism of order 3 having 8 cycles

“…Theorem 2.2 (Huffman [9]) A binary [n, n/2] code C with an automorphism σ defined in (1) is self-dual if and only if the following two conditions hold:…”

The binary extremal self-dual codes of lengths 38 and 40

“…In this case, φ(E σ (C) * ) is a Hermitian quaternary [10,5,4] code. Up to equivalence, two Hermitian quaternary [10,5,4] codes exist, namely B 10 and E 10 (see [19]). The generator matrices of B 10 and E 10 are given by respectively.…”

“…When φ(E σ (C) * ) = 2d 6 , the group L is generated by the permutations (9, 10)(11, 12) , (1, 8, 2, 7) (3,11,6,9,4,12,5,10), (3, 4)(5, 6), (3, 6)(4, 5), (9, 11)(10, 12), (7, 11)(8, 12), and (1, 3)(2, 4). Now we obtain 48 inequivalent self-dual [38, 19,8] codes.…”

“…From [14], we see the extremal self-dual [40, 20,8] codes with a double circulant construction. All binary self-dual [42, 21,8] codes having automorphisms of odd prime orders have been classified in [4,5,7,29,30]. Many binary self-dual [44, 22,8] codes are obtained in [4,5,8,14,25,26,28,30].…”

“…Except for 3-(6,20), 3-(10,8) and 3-(12,2), all codes have been classified. For the extremal [40, 20,8] codes, the possible automorphisms of odd prime order are of types 19-(2,2) (three codes) [27], 7-(5,5) (seven codes) [6,27], 5-(4,20) (two codes) [6,27], 5-(8,0) (82 codes) [6,32], 3-(6,22) (46 codes) [5], 3-(8,16) (49 codes) [18], 3-(10,10), and 3- (12,4). From [14], we see the extremal self-dual [40, 20,8] codes with a double circulant construction.…”