Two-weight and three-weight linear codes based on Weil sums

TL;DR

This paper constructs and analyzes new classes of two-weight and three-weight linear codes using Weil sums, demonstrating their optimality or near-optimality for applications in cryptography and combinatorics.

Contribution

It introduces new classes of linear codes based on Weil sums and determines their weight distributions, some of which are optimal or nearly optimal.

Findings

Several classes of two-weight and three-weight codes are constructed.

The weight distributions of these codes are explicitly determined.

Some codes are proven to be optimal or nearly optimal according to the Griesmer bound.

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 two-weight and three-weight linear codes are presented and their weight distributions are determined using Weil sums. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound.