Self-orthogonality matrix and Reed-Muller code
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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 k and minimum distance d' ≥ d(𝒞). 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 ℛ(2,k). Furthermore, we disprove partially the conjecture (Kim et al. (2021)) by showing that if 31 ≤ n ≤ 256 and n≡ 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 ≤ n ≤ 256 satisfies n 46, 54, 61 and n ≢7, 14, 22, 29, 38, 45, 53, 60 63.
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