This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring R=Fq+uFq+vFq+uvFq, where u2=v2=0,uv=vu, and Fq is a finite field of cardinality q=pr; p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over Fq are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in Fq and Fq+uFq(u2=0) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature.

This paper presents some results on 1-generator quasi-cyclic (QC) codes and their duals over finite fields. We have explicitly obtained a set of generators for the dual of a 1-generator QC code C from the generator of C, for some restricted cases. From the form of the generators of the dual code C⊥, we have derived upper bounds on the minimum distance of C⊥. For their applications in constructing quantum codes, we have presented some criteria for 1-generator QC codes to be self-orthogonal or self-dual. Then using the CSS construction, we have obtained some new and better quantum codes with the help of self-orthogonal 1-generator QC codes.