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\({\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]\)-additive cyclic codes are asymptotically good

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Abstract

We construct a class of \({\mathbb {Z}}_p{\mathbb {Z}}_p[v]\)-additive cyclic codes, where p is a prime number and \(v^2=v\). We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number \(0<\delta <1\) such that the p-ary entropy at \(\frac{k+l}{2}\delta \) is less than \(\frac{1}{2}\), the relative minimum distance of the random code is convergent to \(\delta \) and the rate of the random code is convergent to \(\frac{1}{k+l}\), where p, k, l are pairwise coprime positive integers.

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Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments. This research is supported by the National Natural Science Foundation of China (Nos. 11701336, 11626144, 11671235, 12071264).

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  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, People’s Republic of China

    Xiaotong Hou & Jian Gao

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  1. Xiaotong Hou
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  2. Jian Gao
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Correspondence to Jian Gao.

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Hou, X., Gao, J. \({\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]\)-additive cyclic codes are asymptotically good. J. Appl. Math. Comput. 66, 871–884 (2021). https://doi.org/10.1007/s12190-020-01466-w

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  • DOI: https://doi.org/10.1007/s12190-020-01466-w

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