Self-Orthogonality Matrix and Reed-Muller Codes
Jon-Lark Kim, Whan-Hyuk Choi · IEEE Transactions on Information Theory · 2022
Kim et al. (2021) gave a method to embed a given binary$[n,k]$code$\mathcal {C}\,\,(k = 3, 4)$into a self-orthogonal code of the shortest length which has the same dimension$k$and minimum distance$d' \ge d(\mathcal {C})$. We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix$SO_{k}$, obtained by shortening a Reed-Muller code${\mathcal R}(2,k)$. 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$31 \le n \le 256$and$n\equiv 14,22,29 \pmod {31}$, then there exist optimal$[n], [5]$codes which are self-orthogonal. We also construct optimal self-orthogonal$[n], [6]$codes when$41 \le n \le 256$satisfies$n e 46, 54, 61$and$n \equiv \!\!\!\!\!/~7, 14, 22, 29, 38, 45, 53, 60 \pmod {63}$.