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

In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet F q R , where R = F q + u F q + v F q + u v F q , with u 2 = 1, v 2 = 1, u v = v u and q = p m for odd prime p , positive integer m . Let θ be an automorphism on F q . Extending θ to Θ over R , we study skew ( θ ,Θ)-( λ ,Γ)-constacyclic codes over F q R , where λ and Γ are units in F q and R , respectively. We also show that, the dual of a skew ( θ ,Θ)-( λ ,Γ)-constacyclic code over F q R is a skew ( θ ,Θ)-( λ − 1 ,Γ − 1 )-constacyclic code over 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 F q R . We construct a Gray map over F q R and study the Gray images of skew ( θ ,Θ)-( λ ,Γ)-constacyclic codes over 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 F q .