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\(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes are asymptotically good

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Abstract

We construct a class of \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic code and random quasi-cyclic code of index 2 over \(\mathbb {Z}_{p}\), the asymptotic properties of the rates and relative distances of this class of codes are studied. As a consequence, we prove that \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes are asymptotically good since the asymptotic GV-bound at \(\frac {1+p^{s-1}}{2}\delta \) is greater than \(\frac {1}{2}\), the relative distance of the code is convergent to δ, while the rate is convergent to \(\frac {1}{1+p^{s-1}}\) for \(0< \delta < \frac {1}{1+p^{s-1}}\).

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Correspondence to Shixin Zhu.

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This research is supported by the National Natural Science Foundation of China (No.61772168; No.61572168).

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Yao, T., Zhu, S. \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes are asymptotically good. Cryptogr. Commun. 12, 253–264 (2020). https://doi.org/10.1007/s12095-019-00397-z

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