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.

翻译：本文旨在达成两个目标。首先，我们利用Solomon-Stiffler码及相关剩余码，刻画了达到Griesmer界的二元自正交码的存在性。其次，基于此刻画，我们确定了除五种特例外$d_{so}(n,7)$的精确值，以及除41种特例外$d_{so}(n,8)$的精确值，其中$d_{so}(n,k)$表示所有二元自正交$[n,k]$码的最大最小距离。目前，Shi等人（2022）已确定了$d_{so}(n,k)$（$k \le 6$）的精确值。此外，我们通过分析二元自正交码的剩余码，发展了一种证明某些二元自正交码不存在性的通用方法。