A construction of minimal linear codes from partial difference sets

TL;DR

This paper constructs new minimal linear codes using partial difference sets, providing a character-theoretical characterization and exploring applications in secret sharing schemes.

Contribution

It introduces a novel construction of minimal linear codes from partial difference sets and characterizes their minimality conditions.

Findings

New three-weight and four-weight minimal linear codes

Codes that do not satisfy Ashikhmin-Barg condition

Applications in secret sharing schemes

Abstract

In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes.