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Three new classes of optimal quinary cyclic codes with minimum distance four

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an important subject of study for many years. Recently, several classes of optimal quinary cyclic codes of the forms \(\mathcal {C}_{(0,1,e)}\) and \(\mathcal {C}_{(1,e,s)}\) are presented in the literature, where \(s=\frac{5^m-1}{2}\) and \(2 \le e \le 5^{m}-2\). In this paper, by considering the solutions of certain equations over finite fields, we give three new classes of infinite families of optimal quinary cyclic codes of the form \(\mathcal {C}_{(1,e,s)}\) with parameters \([5^{m}-1, 5^{m}-2m-2, 4]\). Specifically, we make progress towards an open problem proposed by Gaofei Wu et al. [17].

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Authors and Affiliations

  1. College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224003, China

    Yan Liu

  2. School of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiwang Cao

  3. Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing, 210016, China

    Xiwang Cao

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  1. Yan Liu
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Correspondence to Yan Liu.

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I declare that the authors have no competing interests or other interests that might be perceived to influence the results and/or discussion reported in this paper.

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Y. Liu is supported by the National Natural Science Foundation of China (No. 12001475), Natural Science Foundation of Jiangsu Province (No. BK20201059) X. Cao is supported by the National Natural Science Foundation of China (No. 12171241).

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Liu, Y., Cao, X. Three new classes of optimal quinary cyclic codes with minimum distance four. AAECC (2023). https://doi.org/10.1007/s00200-023-00621-7

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