On the additive cyclic structure of quasi-cyclic codes
Güneri, Cem and Özdemir, Funda and Sole, Patrick (2018) On the additive cyclic structure of quasi-cyclic codes. Discrete Mathematics, 341 (10). pp. 2735-2741. ISSN 0012-365X (Print) 1872-681X (Online)
Official URL: http://dx.doi.org/10.1016/j.disc.2018.06.025
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)  to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017)  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.
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