Two Conjectures on the Largest Minimum Distances of Binary Self-Orthogonal Codes With Dimension 5
Minjia Shi, Shitao Li, Jon-Lark Kim · IEEE Transactions on Information Theory · 2023
The purpose of this paper is to solve the two conjectures on the largest minimum distance$d_{so}(n,5)$of a binary self-orthogonal$[n, 5]$code proposed by Kim and Choi (2022). The determination of$d_{so}(n,k)$has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension$k$increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal$[n,k]$codes were constructed for$k=4,5$. Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on$d_{so}(n,5)$. In this paper, we develop a general method to determine the exact value of$d_{so}(n,k)$for$k=5,6$and show that the two conjectures made by Kim and Choi (2022) are true.