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 $\mathbb {F}_{4}$.