Construction of Additive Complementary Dual Codes Over Finite Fields

TL;DR

This paper explores the construction of additive complementary dual (ACD) codes over finite fields, providing new methods to generate codes with improved parameters for applications in coding theory.

Contribution

It introduces novel construction techniques for ACD codes over finite fields using generator matrices and trace inner products, enhancing code parameters over known codes.

Findings

Constructed numerous ACD codes with better parameters than existing codes over F_9 and F_4.

Established methods to derive ACD codes from linear codes over smaller fields.

Provided explicit constructions for ACD codes using trace Hermitian and Euclidean inner products.

Abstract

In this work, we investigate additive complementary dual (ACD) codes and their construction over finite fields $\mathbb{F}_{q^2}$ with respect to the trace inner products, where $q$ is a prime power. First, we associate an additive code with a matrix known as a generator matrix. After that, we describe ACD codes in terms of generator matrices for the trace Hermitian and the trace Euclidean inner products. We also construct ACD codes over $\mathbb{F}_{q^2}$ from linear codes over $\mathbb{F}_q.$ Additionally, we present techniques for constructing ACD codes with various parameters from a given ACD code over $\mathbb{F}_{q^2}.$ By applying these methods, we construct numbers of trace Euclidean and trace Hermitian ACD codes that exhibit better parameters compared to the best known linear codes over $\mathbb{F}_9$ and $\mathbb{F}_4$ of the same size and length.