Asymptotically good $ \mathbb{Z}_{p}\mathbb{Z}_{p}[u]/\langle u^{t}\rangle $-additive cyclic codes

Ting Yao; Heqian Xu; Yongsheng Tang; Shixin Zhu

doi:10.3934/amc.2022087

Article Contents

2023, Volume 17, Issue 5: 1290-1301. Doi: 10.3934/amc.2022087

This issue Previous Article A survey on functional encryption Next Article

Asymptotically good $ \mathbb{Z}_{p}\mathbb{Z}_{p}[u]/\langle u^{t}\rangle $-additive cyclic codes

1.
School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, Anhui, China
2.
School of Mathematics, Hefei University of Technology, Hefei 230601, Anhui, China

^*Corresponding author: Ting Yao
^*Corresponding author: Ting Yao

Received: July 1, 2022

Revised: October 1, 2022

Early access: November 7, 2022

Published online: October 1, 2023

This research is supported by the Natural Science Foundation of Anhui Province (No.2108085QA03); the National Natural Science Foundation of China (Nos.12201170, 12171134); the Natural Science Foundation for the Higher Education Institutions of Anhui Province (No. KJ2021A0926).

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Abstract

$ \mathbb{Z}_{2}(\mathbb{Z}_{2}+u\mathbb{Z}_{2}) $-additive cyclic codes were proved to be asymptotically good in 2020 by Yao et al., where $ u^{2} = 0 $. We extend the study to the double cyclic codes over two finite commutative chain rings. Let $ R_{t} = \mathbb{Z}_{p}[u]/\langle u^{t}\rangle = \mathbb{Z}_{p}+u\mathbb{Z}_{p}+u^{2}\mathbb{Z}_{p}+\ldots+u^{t-1}\mathbb{Z}_{p} $ be a chain ring, where $ u^{t} = 0 $. We construct a class of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes generated by pairs of polynomials, where $ p $ is a prime number. By using probabilistic methods, we study the asymptotic behaviour of the rates and relative minimum distances of a certain class of the codes. We show that there exists an asymptotically good infinite sequence of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $, and the rate is convergent to $ \frac{1}{1+p^{t-1}} $ for $ 0< \delta< \frac{1}{1+p^{t-1}} $, and $ t\geq1 $.

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Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.

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