# A Construction of Linear Codes and Their Complete Weight Enumerators

## TL;DR

This paper constructs a new class of linear codes using defining sets based on trace functions over finite fields, explicitly determines their weight enumerators, and identifies several optimal codes with potential applications in cryptography and combinatorics.

## Contribution

It introduces a novel construction of linear codes from specific trace-based defining sets and explicitly computes their complete weight enumerators using cyclotomic numbers and Gauss sums.

## Key findings

- Explicit formulas for weight enumerators of the constructed codes
- Identification of several optimal codes with few weights
- Codes have higher rates suitable for cryptographic applications

## Abstract

Recently, linear codes constructed from defining sets have been studied extensively. They may have nice parameters if the defining set is chosen properly. Let $ m >2$ be a positive integer. For an odd prime $ p $, let $ r=p^m $ and $\text{Tr}$ be the absolute trace function from $\mathbb{F}_r$ onto $\mathbb{F}_p$. In this paper, we give a construction of linear codes by defining the code $ C_{D}=\{(\mathrm{Tr}(ax))_{x\in D}: a \in \mathbb{F}_{r} \}, $ where $ D =\left\{x\in \mathbb{F}_{r} : \mathrm{Tr}(x)=1, \mathrm{Tr}(x^2)=0 \right\}. $ Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. In addition, we obtain several optimal linear codes with a few weights. They have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes.