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 F-ql via a basis of the extension F-ql/F-q. However, this cyclic code is only linear over F-q, making it an additive cyclic code, or an F-q-linear cyclic code, over the alphabet F-ql. 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 F-ql-linear cyclic images under a basis of the extension F-ql/F-q. 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. (C) 2018 Elsevier B.V. All rights reserved.