Optimal binary codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$-additive cyclic codes

Abstract

In this paper, we study the algebraic structure of additive cyclic codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over $$\mathbb {F} _{2}.$$ Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ -additive cyclic code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic code. Using the mapping W, we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over $$\mathbb {F}_{4}$$ and give some examples of optimal parameter quantum codes over $$\mathbb {F}_{4}$$ .