A New Constructions of Minimal Binary Linear Codes

TL;DR

This paper introduces a new generic construction method for minimal binary linear codes with dimension m+2, providing conditions for minimality, and discovers a class of codes that violate the Ashikhmin-Barg condition with their weight distributions determined.

Contribution

It presents a novel generic construction for minimal binary linear codes and characterizes conditions for minimality, including codes that violate existing conditions.

Findings

New class of minimal binary linear codes violating Ashikhmin-Barg condition

Derived necessary and sufficient conditions for minimality

Determined weight enumerators of the new codes

Abstract

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.