Some results on BCH codes of length $ \frac{{{q^m} + 1}}{2} $

Jiayuan Zhang; Ping Li; Xiaoshan Kai; Zhonghua Sun

doi:10.3934/amc.2023010

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2024, Volume 18, Issue 6: 1605-1621. Doi: 10.3934/amc.2023010

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Some results on BCH codes of length $ \frac{{{q^m} + 1}}{2} $

1.
School of Mathematics, Hefei University of Technology, Hefei 23061, Anhui, China

^*Corresponding author: Ping Li
^*Corresponding author: Ping Li

Received: December 2022

Revised: March 2023

Early access: March 2023

Published: December 2024

This study is supported by the National Natural Science Foundation of China (No.61972126, No.62002093, No.U21A20428, No.12171134)

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Abstract

BCH codes are an especially important kind of cyclic codes, which are extensively used in many domains such as communication and storage systems. Determining the dimensions of BCH codes is a challenging problem in coding theory. In this paper, we study BCH codes of length $ \frac{{{q^m} + 1}}{2} $, for odd $ m $ and $ m \equiv 2(\bmod\; {4}) $. The first few largest coset leaders of such BCH codes are obtained. On the basis of this, we determine their parameters and obtain a few optimal or almost optimal codes.

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Mathematics Subject Classification: Primary: 94B15; Secondary: 94B65.

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References

[1]	P. Charpin, Open problems on cyclic codes, Handbook of Coding Theory, North-Holland, 1 (1998), 963-1063.
[2]	C. Ding, Parameters of several classes of BCH codes, IEEE Trans. Inf. Theory, 61 (2015), 5322-5330. doi: 10.1109/TIT.2015.2470251.
[3]	C. Ding, C. Fan and Z. Zhou, The dimension and minimum distance of two classes of primitive BCH codes, Finite Fields Appl., 45 (2017), 237-263. doi: 10.1016/j.ffa.2016.12.009.
[4]	D. Gorenstein and N. Zierler, A class of error-correcting codes in ${p^m}$ symbols, J. Soc. Ind. Appl. Math., 9 (1961), 207-214. doi: 10.1109/TIT.2017.2672961.
[5]	A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.
[6]	C. Li, C. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 344-4356. doi: 10.1109/TIT.2017.2672961.
[7]	S. Li, C. Ding, M. Xiong and G. Ge, Narrow-sense BCH codes over ${\mathrm {GF}}(q) $ with length $ n = \frac {q^{m}-1}{q-1} $, IEEE Trans. Inf. Theory, 63 (2017), 7219-7236. doi: 10.1109/tit.2017.2743687.
[8]	S. Li, C. Li, C. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 (2017), 5699-5717. doi: 10.1109/tit.2017.2723363.
[9]	X. Ling, S. Mesnager, Y. Qi and C. Tang, A class of narrow-sense BCH codes over $\mathbb {F} _q $ of length $\frac {q^ m-1}{2} $, Des. Codes Cryptogr., 88 (2020), 413-427. doi: 10.1007/s10623-019-00691-0.
[10]	H. Liu, C. Ding and C. Li, Dimensions of three types of BCH codes over $GF(q)$, Discrete Math., 340 (2017), 1910-1927. doi: 10.1016/j.disc.2017.04.001.
[11]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977.
[12]	L. Sok, M. Shi and P. Solé, Construction of optimal LCD codes over large finite fields, Finite Fields Appl., 50 (2018), 138-153. doi: 10.1016/j.ffa.2017.11.007.
[13]	M. Shi, D. Huang, L. Sok and P. Solé, Double circulant LCD codes over $\mathbb{Z}_4$, Finite Fields Appl., 58 (2019), 133-144. doi: 10.1016/j.ffa.2019.04.001.
[14]	M. Shi, F. Özbudak, X. Li and P. Solé, LCD codes from tridiagonal Toeplitz matrices, Finite Fields Appl., 75 (2021), 101892. doi: 10.1016/j.ffa.2021.101892.
[15]	H. Yan, H. Liu, C. Li and S. Yang, Parameters of LCD BCH codes with two lengths, Adv. Math. Commun., 12 (2018), 579-594.
[16]	X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393. doi: 10.1016/0012-365x(94)90283-6.
[17]	H. Zhu, M. Shi, X. Wang and T. Helleseth, The $q$-ary antiprimitive BCH codes, IEEE Trans. Inf. Theory, 68 (2022), 1683-1695. doi: 10.1109/TIT.2021.3131810.
[18]	S. Zhu, Z. Sun and X. Kai, A class of narrow-sense BCH codes, IEEE Trans. Inf. Theory, 65 (2019), 4699-4714. doi: 10.1109/tit.2019.2913389.