Two conjectures on the largest minimum distances of binary self-orthogonal codes with dimension 5

TL;DR

This paper proves two conjectures regarding the maximum minimum distance of binary self-orthogonal codes with dimension 5, advancing understanding of their optimal parameters.

Contribution

It develops a general method to determine the exact maximum minimum distance for dimension 5 and 6, confirming two previously proposed conjectures.

Findings

Confirmed the two conjectures on $d_{so}(n,5)$

Established a method for exact determination of $d_{so}(n,5)$ and $d_{so}(n,6)$

Enhanced understanding of binary self-orthogonal code parameters

Abstract

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 (IEEE Trans. Inf. Theory, 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.