A construction of $q$-ary linear codes with two weights

TL;DR

This paper introduces a new method for constructing $q$-ary linear codes with two weights using trace and norm functions, resulting in optimal or near-optimal codes with applications in secret sharing and graph theory.

Contribution

The paper presents a novel construction of $q$-ary linear codes with two weights based on trace and norm functions, expanding the class of known optimal codes.

Findings

Codes can be optimal or almost optimal.

Codes can be used to construct secret sharing schemes.

Codes lead to strongly regular graphs with new parameters.

Abstract

Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of $q$-ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters.