Construction of Additive Complementary Dual Codes Over Finite Fields
Gyanendra K. Verma, R. K. Sharma
TL;DR
The paper addresses the construction of additive complementary dual (ACD) codes over $\mathbb{F}_{q^2}$ with respect to trace inner products, linking them to LCD codes and duality concepts. It develops generator-matrix criteria: a trace Hermitian ACD code is obtained when $G\overline{G}^T+\overline{G}G^T$ is invertible, while a trace Euclidean ACD code requires $G G^T+\overline{G}\overline{G}^T$ to be invertible, together with explicit trace-based projection maps. It then provides methods to construct ACD codes from LCD codes over $\mathbb{F}_q$ via $E=aC+bD$ with trace-orthogonality properties, and presents recursive extension techniques that add coordinates or scale by elements in $C^{\perp}$ to obtain $(n+1,q^{k+1})_{q^2}$ or $(n,q^{k+1})_{q^2}$ codes. Extensive computational experiments using MAGMA yield multiple new trace Euclidean and trace Hermitian ACD codes with parameters surpassing the best known linear codes over $\mathbb{F}_9$ and $\mathbb{F}_4$, demonstrating the practical impact of the methods on small-field code design for potential quantum and security applications.
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.