Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes

TL;DR

This paper introduces a method to construct minimal linear codes that violate the Ashikhmin-Barg condition from arbitrary projective linear codes, providing numerous families with near-optimal minimum weights and self-orthogonality.

Contribution

It presents a general transformation technique from projective linear codes to minimal codes violating the Ashikhmin-Barg condition and constructs many such codes with detailed weight distributions.

Findings

Constructed infinite families of minimal codes violating the Ashikhmin-Barg condition.

Determined weight distributions of the constructed codes.

Produced codes with minimum weights close to optimal or best known.

Abstract

In recent years, there have been many constructions of minimal linear codes violating the Ashikhmin-Barg condition from Boolean functions, linear codes with few nonzero weights or partial difference sets. In this paper, we first give a general method to transform a minimal code satisfying the Ashikhmin-Barg condition to a minimal code violating the Ashikhmin-Barg condition. Then we give a construction of a minimal code satisfying the Ashikhmin-Barg condition from an arbitrary projective linear code. Hence an arbitrary projective linear code can be transformed to a minimal codes violating the Ashikhmin-Barg condition. Then we give infinite many families of minimal codes violating the Ashikhamin-Barg condition. Weight distributions of constructed minimal codes violating the Ashikhmin-Barg condition in this paper are determined. Many minimal linear codes violating the Ashikhmin-Barg condition with their minimum weights close to the optimal or the best known minimum weights of linear codes are constructed in this paper. Moreover, many infinite families of self-orthogonal binary minimal codes violating the Ashikhmin-Barg condition are also given.