$\mathbb{F}_q\mathbb{F}_{q^2}$-additive cyclic codes and their Gray images
Ankit Yadav, Ritumoni Sarma
TL;DR
This work develops the theory of additive cyclic codes over the mixed alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, deriving explicit generator structures and minimal spanning sets, and analyzing duals within a unified polynomial-module framework. Through a Gray map, the authors translate these mixed-alphabet codes into $\mathbb{F}_{q}$-linear codes, obtaining quasi-cyclic and cyclic codes with strong distance properties, including several optimal ternary codes. The paper also establishes conditions under which the Gray images are LCD, and demonstrates the construction of several optimal ternary LCD codes from $\mathbb{F}_{3}\mathbb{F}_{9}$-additive codes. Together, these results extend additive code theory to mixed alphabets, produce new good codes relevant for classical and quantum applications, and suggest avenues for further exploration of asymptotic behavior and higher-tower alphabets. The approach combines explicit generator descriptions, minimal spanning sets, duality, and distance-preserving Gray images to yield practically useful codes with provable optimality indicators.
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.