  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.
