The concatenated structure of quasi-cyclic codes and an improvement of Jensen's bound

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Official URL: http://dx.doi.org/10.1109/TIT.2012.2225823

Following Jensen's work from 1985 a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Sole. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constituent codes are two-dimensional cyclic codes. In fact, we obtain a generalization of this result to multidimensional cyclic codes. The concatenated structure also yields a lower bound on the minimum distance of quasi-cyclic codes, as noted by Jensen, which we call Jensen's bound. We show that a recent lower bound on the minimum distance of quasi-cyclic codes that we obtained is in general better than Jensen's lower bound.