A Class of Linear Codes with a Few Weights

TL;DR

This paper introduces a new class of linear codes over finite fields with few weights, determines their weight distributions, and identifies some as optimal codes meeting specific bounds.

Contribution

The paper presents a novel class of linear codes with few weights over finite fields and determines their weight distributions, including some optimal codes.

Findings

Identified a new class of linear codes with few weights.

Determined the weight distributions of these codes.

Found some codes that are optimal, meeting certain bounds.

Abstract

Linear codes have been an interesting subject of study for many years, as linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a class of linear codes with a few weights over the finite field $\gf(p)$ are presented and their weight distributions are also determined, where $p$ is an odd prime. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes.