Two Conjectures on the Largest Minimum Distances of Binary Self-Orthogonal Codes With Dimension 5

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.