Generalized Quasi-Cyclic Codes with Arbitrary Block Lengths

This paper considers generalized quasi-cyclic (GQC) codes with no restriction on their block lengths. By relaxing the condition on its block lengths, we find some new optimal codes of small length. Also, we generalize the decomposition of codes and dimension formula given by Güneri et al, Séguin, and Siap et al. We use two different approaches to describe GQC codes: first , by torsion module structure, and second , by injection into classes of QC codes. Using the first approach, we can determine the dimension and give a formula for the minimum distance of the corresponding GQC code. In the second approach, we use structural properties of QC codes with one specific length. This approach gives us a way to construct GQC codes, dual code for a given GQC code, and a criterion for a GQC code to be a Euclidean self-dual code. Moreover, we also consider GQC codes over a family of finite rings, called the ring B k , and its relation to GQC codes over finite fields using a collection of Gray maps.