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 FqR, where R= Fq+ uFq+ vFq+ uvFq, with u2 = 1,v2 = 1, uv = vu and q = pm for odd prime p, positive integer m. Let θ be an automorphism on Fq. Extending θ to Θ over R, we study skew (θ,Θ)-(λ,Γ)-constacyclic codes over FqR, where λ and Γ are units in Fq and R, respectively. We also show that, the dual of a skew (θ,Θ)-(λ,Γ)-constacyclic code over FqR is a skew (θ,Θ)-(λ− 1,Γ− 1)-constacyclic code over FqR. 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 FqR. We construct a Gray map over FqR and study the Gray images of skew (θ,Θ)-(λ,Γ)-constacyclic codes over FqR. 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 Fq.