(β,γ)-Skew QC Codes with Derivation over a Semi-Local Ring

In this article, we consider a semi-local ring S=F-q+uF(q), where u2=u, q=p(s )and p is a prime number. We define a multiplication yb=beta(b)y+gamma(b), where beta is an automorphism and gamma is a beta-derivation on S so that S[y;beta,gamma] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;beta,gamma] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.