Abstract
Quasi-cyclic codes are generalizations of the familiar linear cyclic codes. By using the results of [4], the authors in [2, 3] showed that a quasi-cyclic code \(\mathscr {C}\) over \(\mathbb {F}_q\) of length \(\ell m\) and index \(\ell \) with m being pairwise coprime to \(\ell \) and the characteristic of \(\mathbb {F}_q\) is equivalent to a cyclic code if the constituent codes of \(\mathscr {C}\) are cyclic, where q is a prime power and the equivalence is given in [3]. In this paper, we apply an algebraic method to prove that a quasi-cyclic code with cyclic constituent codes is equivalent to a cyclic code. Moreover, the main result (see Theorem 4) includes Proposition 9 in [3] as a special case.
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References
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Acknowledgements
This research is supported by National Natural Science Foundation of China (61202068, 61672036 and 11526045), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key Projects of Support Program for outstanding young talents in Colleges and Universities (gxyqZD2016008).
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Shi, M., Zhang, Y., Solé, P. (2017). Notes on Quasi-Cyclic Codes with Cyclic Constituent Codes. In: Abualrub, T., Jarrah, A., Kallel, S., Sulieman, H. (eds) Mathematics Across Contemporary Sciences. AUS-ICMS 2015. Springer Proceedings in Mathematics & Statistics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-46310-0_12
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DOI: https://doi.org/10.1007/978-3-319-46310-0_12
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