Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances

Abstract

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing the Solomon-Stiffler codes. As a result, we reduce a problem with an infinite number of cases to a finite number of cases. Second, 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. Using such a characterization, we completely determine the exact value of $d_{so}(n,7)$, where $d_{so}(n,k)$ denotes the largest minimum distance among all binary self-orthogonal $[n,k]$ codes.