Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

Abstract

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56, 12, 65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56, 28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56, 12, 65) designs.

Skew Hadamard designs and their codes

“…Similarly there is a unique SH matrix of order 12 [9], whose C(A) is the ternary Golay [12,6,6] code. These can be explained from Proposition 3.…”

“…So our codes are inequivalent to these codes. Later, Harada [6] constructed at least 1135 Type II [56,26,12] codes from self-orthogonal 3-(56, 12, 65) designs. It will be interesting to check the equivalence of our codes with his codes.…”

3-(2m, 2m–1, 2m–2–1) design for first order Reed-Muller code R(1,m)

“…Out of these 2(= 2 3−1 − 2) rows, 1 row belongs to north-west part of C(1, 3) and remaining 1 row belongs to south-west part of C(1, 3) . If we choose any one column from last 4 columns of C (1, 3) , we get right side of corresponding two consecutive 1's, one position is 1 and one position is 0, since south-east part of C (1, 3) is the complement of north-east part of C (1,3) . Hence we get only one row, which contains three consecutive 1's.…”

“…A self-dual codes C is doubly-even if all code words have weights divisible by 4. Such codes exist [1] if and only if n ≡ 0 (mod 8). The minimum weight d of a doubly-even self-dual code of length n is upper bounded by d ≤ 4[n/24] + 4 .…”

“…A t-(n, w, λ) design D is a set X of n points together with a collection of w -subsets of X (called block) such that every t -subset of X is contained in exactly λ blocks [1].…”