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

TL;DR

This paper characterizes the existence of binary self-orthogonal codes meeting the Griesmer bound, determines their maximum minimum distances for certain parameters, and develops methods to prove nonexistence of some codes.

Contribution

It provides a characterization of binary self-orthogonal codes meeting the Griesmer bound and determines exact maximum minimum distances for specific parameters.

Findings

Characterization of codes meeting the Griesmer bound using Solomon-Stiffler codes

Exact values of $d_{so}(n,7)$ and $d_{so}(n,8)$ for most cases

A general method for proving nonexistence of certain binary self-orthogonal codes

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 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.