Abstract
Let \({\mathbb {F}}_{2}[u]={\mathbb {F}}_{2}+u{\mathbb {F}}_{2}\), \(u^2=0\). In this paper, we construct a class of \({\mathbb {F}}_{2}[u]{\mathbb {F}}_{2}[u]\)-additive cyclic codes generated by pairs of polynomials. We discuss their algebraic structure and show that generator matrices can be obtained 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 and k and l be co-prime odd positive integers such that the entropy \(h_{2}(\frac{(k+l)\delta }{4})<\frac{1}{2},\) we show that the relative minimum distance converges to \(\delta \) and the rates of the random codes converge to \(\frac{1}{k+l}\). Finally, we conclude that the \({\mathbb {F}}_{2}[u]{\mathbb {F}}_{2}[u] \)-additive cyclic codes are asymptotically good and provide some examples for this class of codes.
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Acknowledgements
B.P. Yadav wants to thank CSIR for its financial support through file No. 09/1023(0018)/2016 EMR-I, and A.K. Upadhyay thanks SERB DST for their support through project MTR/2020/000006. A part of this paper was written during a stay of H.Q. Dinh and B.P. Yadav in the Vietnam Institute For Advanced Study in Mathematics (VIASM) in Summer 2022, they 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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Dinh, H.Q., Yadav, B.P., Pathak, S. et al. \({\mathbb {F}}_{2}[u]{\mathbb {F}}_{2}[u] \)-additive cyclic codes are asymptotically good. J. Appl. Math. Comput. 69, 1037–1056 (2023). https://doi.org/10.1007/s12190-022-01771-6
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DOI: https://doi.org/10.1007/s12190-022-01771-6