  The purpose of this paper is two-fold. First, we characterize the existence
of binary self-orthogonal codes meeting the Griesmer bound by employing
Solomon-Stiffler codes and some related residual codes. Second, using such a
characterization, we determine the exact value of $d_{so}(n,7)$ except for five
special cases and the exact value of $d_{so}(n,8)$ except for 41 special cases,
where $d_{so}(n,k)$ denotes the largest minimum distance among all binary
self-orthogonal $[n, k]$ codes. Currently, the exact value of $d_{so}(n,k)$ $(k
\le 6)$ was determined by Shi et al. (2022). In addition, we develop a general
method to prove the nonexistence of some binary self-orthogonal codes by
considering the residual code of a binary self-orthogonal code.
