Self-orthogonality matrix and Reed-Muller code

Abstract

Kim et al. (2021) gave a method to embed a given binary [ , ] code C ( = 3, 4) into a self-orthogonal code of the shortest length which has the same dimension and minimum distance ′ ≥ (C). We extends this result for = 5 and 6 by proposing a new method related to a special matrix, called the self-orthogonality matrix , obtained by shortnening a Reed-Muller code R (2, ). Furthermore, we disprove partially the conjecture (Kim et al. (2021)) by showing that if 31 ≤ ≤ 256 and ≡ 14, 22, 29 (mod 31), then there exist optimal [ , 5] codes which are self-orthogonal. We also construct optimal self-orthogonal [ , 6] codes when 41 ≤ ≤ 256 satisfies