On the additive cyclic structure of quasi-cyclic codes

摘要

An index?, lengthm?quasi-cyclic code can be viewed as a cyclic code of lengthmover the fieldFq?via a basis of the extensionFq?∕Fq. However, this cyclic code is only linear overFq, making it an additive cyclic code, or anFq-linear cyclic code, over the alphabetFq?. 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 haveFq?-linear cyclic images under a basis of the extensionFq?∕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.