Constructions of binary self-orthogonal singly-even minimal linear codes violating the Aschikhmin-Barg condition with few weights

TL;DR

This paper introduces new frameworks for constructing binary self-orthogonal singly-even minimal linear codes with few weights, violating the AB condition, and provides their weight distributions and optimality analysis.

Contribution

It establishes a complete characterization of such codes, develops three construction frameworks, and generates many infinite classes with fully determined weight distributions.

Findings

Many infinite classes of codes with few weights were constructed.

Some dual codes are optimal or near-optimal.

The paper provides necessary and sufficient conditions for code construction.

Abstract

We first establish a simple yet powerful necessary and sufficient condition for a binary linear code to be SO, leading to a complete characterization of singly-even codes in this family. We further derive necessary and sufficient conditions on Boolean and vectorial Boolean functions for generating such codes via a standard construction method. Building on this foundation, we propose three general frameworks for constructing binary SO singly-even minimal non-AB linear codes with few weights. The first two approaches are based on designing Boolean and vectorial Boolean functions that simultaneously satisfy multiple conditions. The third method generates new SO codes from existing ones. As a result, we obtain many infinite classes of binary self-orthogonal singly-even minimal linear codes violating the AB condition with few weights and fully determined weight distributions. Particularly, numerical results show that some duals of our codes are optimal or near-optimal.