Extending binary linear codes to self-orthogonal codes

Abstract

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 $d^{\prime }\ge d(\mathcal{C})$. We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix $S{O}_k$, 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 $n\equiv 14,22,29(\mathrm{mod}31)$, 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 $n\ne 46,54,61$ and $n\not\equiv 7,14,22,29,38,45,53,60(\mathrm{mod}63)$.