Several new classes of self-orthogonal minimal linear codes violating the Ashikhmin-Barg condition

TL;DR

This paper introduces several new classes of self-orthogonal minimal linear codes over finite fields that violate the Ashikhmin-Barg condition, expanding the understanding of code constructions with specific orthogonality and minimality properties.

Contribution

The paper constructs the first known self-orthogonal minimal linear codes that violate the Ashikhmin-Barg condition, including classes over $ extbf{F}_2$ and $ extbf{F}_p$, with explicit weight distributions.

Findings

Constructed self-orthogonal minimal codes over $ extbf{F}_2$ violating AB condition.

Developed classes of self-orthogonal codes from $p$-ary functions with optimal properties.

Provided weight distributions for the constructed codes.

Abstract

Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field $\mathbb{F}_p$. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field $\mathbb{F}_2$ and determine their weight distributions. Next, for any odd prime $p$, we construct two classes of self-orthogonal linear codes from $p$-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.