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：本文的目的是两方面的。首先，我们使用Solomon-Stiffler码和一些相关的余码来表征满足Griesmer界的二元自交错码的存在性。其次，利用这种特征，我们确定了$d_{so}(n,7)$的确切值，除了五种特殊情况和$d_{so}(n,8)$的确切值，除了41种特殊情况，其中$d_{so}(n,k)$表示所有二元自交错$[n, k]$码中最大的最小距离。目前，Shi等人（2022）已经确定了$d_{so}(n,k)$ $(k \le 6)$ 的确切值。此外，我们还开发了一种一般方法，通过考虑二元自交错码的余码来证明某些二元自交错码的不存在性。
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