$\mathbb{F}_q\mathbb{F}_{q^2}$-additive cyclic codes and their Gray images

TL;DR

This paper studies additive cyclic codes over mixed finite fields, determines their generator polynomials, constructs optimal codes satisfying the Singleton bound, and produces optimal linear and LCD codes via Gray maps.

Contribution

It introduces a detailed analysis of additive cyclic codes over $\

Findings

Constructed $\

Produced optimal linear codes over $\

Obtained optimal ternary LCD codes from additive codes

Abstract

We investigate additive cyclic codes over the alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of $\mathbb{F}_{q^2}$-additive cyclic codes that satisfy the well-known Singleton bound are constructed. Using a Gray map, we produce certain optimal linear codes over $\mathbb{F}_{3}$. Finally, we obtain a few optimal ternary linear complementary dual (LCD) codes from $\mathbb{F}_{3}\mathbb{F}_{9}$-additive codes.