On the additive cyclic structure of quasi-cyclic codes

An index ℓ, length mℓ quasi-cyclic code can be viewed as a cyclic code of length m over the field Fqℓ via a basis of the extension Fqℓ∕Fq. However, this cyclic code is only linear over Fq, making it an additive cyclic code, or an Fq-linear cyclic code, over the alphabet Fqℓ. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have Fqℓ-linear cyclic images under a basis of the extension Fqℓ∕Fq. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.