Several classes of $p$-ary linear codes with few-weights derived from Weil sums

TL;DR

This paper introduces new classes of few-weight linear codes over finite fields, constructed via Weil sums and bent functions, with some achieving optimal parameters meeting the Griesmer bound.

Contribution

The paper presents novel constructions of linear codes with few weights using Weil sums and bent functions, expanding the known classes and their parameters.

Findings

Five classes of 4-weight codes were constructed.

Two classes of 6-weight, 8-weight, and 9-weight codes were derived.

An optimal 2-weight code meeting the Griesmer bound was identified.

Abstract

Linear codes with few weights have been a significant area of research in coding theory for many years, due to their applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. Inspired by the works of Cheng and Gao \cite{P8} and Wu, Li and Zeng \cite{P12}, in this paper, we propose several new classes of few-weight linear codes over the finite field $\mathbb{F}_{p}$ through the selection of two specific defining sets. Consequently, we obtain five classes of $4$-weight linear codes and one class of $2$-weight linear codes from our first defining set. Furthermore, by employing weakly regular bent functions in our second defining set, we derive two classes of $6$-weight codes, two classes of $8$-weight codes, and one class of $9$-weight codes. The parameters and weight distributions of all these constructed codes are wholly determined by detailed calculations on certain Weil sums over finite fields. In addition, we identify an optimal class of $2$-weight codes that meet the Griesmer bound.