Some three-weight linear codes and their complete weight enumerators and weight hierarchies

TL;DR

This paper constructs a new class of three-weight linear codes over finite fields using quadratic functions, and fully determines their weight enumerators and hierarchies, extending previous research in the area.

Contribution

It introduces a bivariate construction method for three-weight codes and provides complete weight enumerators and hierarchies, generalizing earlier results.

Findings

Constructed new three-weight linear codes from quadratic functions.

Determined complete weight enumerators of the codes.

Established the weight hierarchies of the codes.

Abstract

Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power $q$, we construct a class of three-weight $\F_q$-linear codes from quadratic functions via a bivariate construction and then determine the complete weight enumerators and weight hierarchies of these linear codes completely. This paper generalizes some results in Li et al. (2022) and Hu et al. (2024).