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 FqR, where R = F-q + uF(q) + vF(q) + uvF(q), with u(2) = 1,v(2) = 1, uv = vu and q = p(m) for odd prime p, positive integer m. Let theta be an automorphism on F-q. Extending theta to Theta over R, we study skew (theta, Theta)-(lambda,Gamma)-constacyclic codes over F-q R, where lambda and G are units in F-q and R, respectively. We also show that, the dual of a skew (theta, Theta)-(lambda,Gamma)-constacyclic code over FqR is a skew (theta, Theta)-(lambda(-1),Gamma(-1))-constacyclic code over FqR. We classify some self-dual skew(theta, Theta)-(lambda,Gamma)-constacyclic codes using the possible values of units of R. Also using suitable values of lambda,theta,Gamma and Theta, we present the structure of other linear codes over FqR. We construct a Gray map over FqR and study the Gray images of skew (theta, Theta)-(lambda,Gamma)-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 theta-cyclic codes than from cyclic codes over F-q.

Automated summary

In this paper, the authors first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet FqR. Then, skew (theta, Theta)-(lambda,Gamma)-constacyclic codes over FqR are studied and their dual codes are classified. The authors also construct a Gray map over FqR and study the Gray images of skew (theta, Theta)-(lambda,Gamma)-constacyclic codes. Many good codes, including optimal codes and near-optimal codes, are constructed, and the advantages of using skew cyclic codes in the construction of quantum error-correcting codes (QECCs) are discussed.