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\(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good

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Abstract

In this paper, we construct a class of \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes generated by 3-tuples of polynomials. We discuss their algebraic structure and show that generator matrices can be constructed for all codes in this class. We study asymptotic properties of this class of codes by using a Bernoulli random variable. Moreover, let \(0< \delta < 1\) be a real number such that the entropy \(h_{4}(\frac{(k+l+t)\delta }{6})<\frac{1}{4},\) we show that the relative minimum distance converges to \(\delta\) and the rate of the random codes converges to \(\frac{1}{k+l+t},\) where k, l, and t are pairwise co-prime positive odd integers. Finally, we conclude that the \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good.

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Acknowledgements

The authors would like to thank the editor and anonymous referees for their valuable suggestions and comments which have highly improved this paper. BPY wants to thank CSIR for its financial support through (file No. 09/1023(0018)/2016 EMR-I) and AKU thanks SERB DST for their support through project (MTR/2020/000006). S. Pathak acknowledges the research support by iHub-Anubhuti-IIITD Foundation set up under the NM-ICPS scheme of the Department of Science and Technology, India, under Grant no. IHUB Anubhuti/Project Grant/12. A part of this paper was written during a stay of H.Q. Dinh in the Vietnam Institute For Advanced Study in Mathematics (VIASM) in Summer 2022, he would like to thank the members of VIASM for their hospitality. This paper is partially supported by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand.

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  1. Sachin Pathak

    Present address: Department of Mathematics, IIIT Delhi, New Delhi, 110020, India

Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Warren, OH, 44483, USA

    Hai Q. Dinh

  2. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801103, India

    Bhanu Pratap Yadav

  3. Department of Mathematics, SRAP College, B R A Bihar University, Muzaffarpur, Bihar, 842001, India

    Abhyendra Prasad

  4. Department of Mathematics, Banaras Hindu University, Varanasi, Uttar Pradesh, 221005, India

    Ashish Kumar Upadhyay

  5. Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai, 52000, Thailand

    Woraphon Yamaka

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  1. Hai Q. Dinh
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Dinh, H.Q., Yadav, B.P., Pathak, S. et al. \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good. AAECC 35, 485–505 (2024). https://doi.org/10.1007/s00200-022-00557-4

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