Several classes of linear codes with few weights derived from Weil sums

TL;DR

This paper constructs several classes of linear codes with few weights using Weil sums, providing their parameters and weight distributions, and identifies some optimal codes meeting the Griesmer bound.

Contribution

It introduces new classes of multi-weight linear codes derived from Weil sums with explicit parameters and optimality properties.

Findings

Explicit weight distributions for the constructed codes

Identification of three classes of optimal codes meeting the Griesmer bound

Many (almost) optimal codes can be obtained from the new constructions

Abstract

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with the defining sets given by the intersection, difference and union of two certain sets, where $t=3,4,5,6$ and $q$ is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.