On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes

Abstract

In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet $\mathbb {F}_{q}R$, where $R=\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}$, with u² = 1,v² = 1, uv = vu and q = p^m for odd prime p, positive integer m. Let 𝜃 be an automorphism on $\mathbb {F}_{q}$. Extending 𝜃 to Θ over R, we study skew (𝜃,Θ)-(λ,Γ)-constacyclic codes over $\mathbb {F}_{q}R$, where λ and Γ are units in $\mathbb {F}_{q}$ and R, respectively. We also show that, the dual of a skew (𝜃,Θ)-(λ,Γ)-constacyclic code over $\mathbb {F}_{q}R$ is a skew (𝜃,Θ)-(λ^− 1,Γ^− 1)-constacyclic code over $\mathbb {F}_{q}R$. We classify some self-dual skew (𝜃,Θ)-(λ,Γ)-constacyclic codes using the possible values of units of R. Also using suitable values of λ,𝜃,Γ and Θ, we present the structure of other linear codes over $\mathbb {F}_{q} R$. We construct a Gray map over $\mathbb {F}_{q}R$ and study the Gray images of skew (𝜃,Θ)-(λ,Γ)-constacyclic codes over $\mathbb {F}_{q}R$. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew 𝜃-cyclic codes than from cyclic codes over $\mathbb {F}_{q}$.