Self-orthogonality matrix and Reed-Muller code

11/24/2021
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by   Jon-Lark Kim, et al.
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Kim et al. (2021) gave a method to embed a given binary [n,k] code š’ž (k = 3, 4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ā‰„ d(š’ž). We extends this result for k=5 and 6 by proposing a new method related to a special matrix, called the self-orthogonality matrix SO_k, obtained by shortnening a Reed-Muller code ā„›(2,k). Furthermore, we disprove partially the conjecture (Kim et al. (2021)) by showing that if 31 ā‰¤ n ā‰¤ 256 and nā‰” 14,22,29 31, then there exist optimal [n,5] codes which are self-orthogonal. We also construct optimal self-orthogonal [n,6] codes when 41 ā‰¤ n ā‰¤ 256 satisfies n 46, 54, 61 and n ā‰¢7, 14, 22, 29, 38, 45, 53, 60 63.

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