Self-Orthogonality Matrix and Reed-Muller Codes

초록

Kim et al. (2021) gave a method to embed a given binary [n, k] code C (k = 3, 4) into a self-orthogonal code of the shortest length which has the same dimension It and minimum distance d' >= d(C). We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix SOk, obtained by shortening a Reed-Muller code R(2, k). Using this approach, we can extend binary linear codes to many optimal self-orthogonal codes of dimensions 5 and 6. Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by showing that if 31 <= n <= 256 and n 14, 22, 29 (mod 31), then there exist optimal [n, 5] codes which are self-orthogonal. We also construct optimal self-orthogonal [n, 6] codes when 41 <= n <= 256 satisfies n not equal 46, 54, 61 and n not equal 7, 14, 22, 29, 38, 45, 53, 60 (mod 63).

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