Complete weight enumerators for several classes of two-weight and three-weight linear codes

TL;DR

This paper constructs new classes of two-weight and three-weight linear codes over finite fields, determines their complete weight enumerators using Weil sums, and identifies some as optimal or near-optimal relative to the Griesmer bound.

Contribution

It extends previous constructions to produce new codes and explicitly computes their weight enumerators, generalizing earlier results.

Findings

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

Complete weight enumerators are explicitly determined for the constructed codes.

The construction generalizes prior work by Li et al. and others.

Abstract

In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in \cite{CL2016, GJ2019}.