Abstract
As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters \([p^m-1, p^m-\frac{3m}{2}-2, 4]\) by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code \(\mathcal {C}_{(0,1,e)}\) and present three classes of optimal codes with parameters \([7^m-1,7^m-2m-2,4]\). Additionally, we provide the weight distribution of one class of their duals.
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References
Charpin, P., Tietäväinen, A., Zinoviev, V.: On the minimum distances of non-binary cyclic codes. Des. Codes Crypt. 17, 81–85 (1999)
Charpin, P., Tietäväinen, A., Zinoviev, V.: On binary cyclic codes with minimum distance d = 3. Problems Inf. Transmiss. 33, 3–14 (1997)
Carlet, C., Ding, C., Yuan, J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005)
Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013)
Li, N., Li, C., Helleseth, T., Ding, C., Tang, X.: Optimal ternary cyclic codes with minimum distance four and five. Finite Fields Appl. 30, 100–120 (2014)
Li, N., Zhou, Z., Helleseth, T.: On a conjecture about a class of optimal ternary cyclic codes. In: 7th International workshop on signal design and its applications in communications, pp. 62-65 (2015)
Wang, L., Wu, G.: Several classes of optimal ternary cyclic codes with minimal distance four. Finite Fields Appl. 40, 126–137 (2016)
Han, D., Yan, H.: On an open problem about a class of optimal ternary cyclic codes. Finite Fields Appl. 59, 335–343 (2019)
Zha, Z., Hu, L.: New classes of optimal ternary cyclic codes with minimum distance four. Finite Fields Appl. 64, 101671 (2020)
Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)
Fan, C., Li, N., Zhou, Z.: A class of optimal ternary cyclic codes and their duals. Finite Fields Appl. 37, 193–202 (2016)
Yan, H., Zhou, Z., Du, X.: A family of optimal ternary cyclic codes from Niho-type exponent. Finite Fiels Appl. 54, 101–112 (2018)
Liu, Q., Liu, X.: On some conjectures about optimal ternary cyclic codes. Appl. Algrbr. Eng. Comm. 33, 419–436 (2022)
Liu, Y., Cao, X., Lu, W.: On some conjectures about optimal ternary cyclic codes. Des. Codes Crypt. 88, 297–309 (2020)
Zha, Z., Hu, L., Liu, Y., Cao, X.: Further results on optimal ternary cyclic codes. Finite Fields Appl. 75(6), 101898 (2021)
Liu, Y., Cao, X., Lu, W.: Two classes of new optimal ternary cyclic codes. Adv. Math. Commun. (2021). https://doi.org/10.3934/amc.2021033
Wang, D., Cao, X.: A family of optimal ternary cyclic codes with minimum distance five and their duals. Crypt. Commun. 14, 1–13 (2022)
Zhao, H., Luo, R., Sun, T.: Two families of optimal ternary cyclic codes with minimal distance four. Finite Fields Appl. 79, 101995 (2022)
Xu, G., Cao, X., Xu, S.: Optimal p-ary cyclic codes with minimum distance four from monomials. Crypt. Commun. 8(4), 541–554 (2016)
Zhou, Y., Kai, X., Zhu, S., Li, J.: On the minimum distance of negacyclic codes with two zeros. Finite Fields Appl. 55, 134–150 (2019)
Liao, D., Kai, X., Zhu, S., Li, P.: A class of optimal cyclic codes with two zeros. IEEE Commun. Lett. 23(8), 1293–1296 (2019)
Liu, Y., Cao, X.: Four classes of optimal quinary cyclic codes. IEEE Comm. Lett. 24(7), 1387–1390 (2020)
Liu, Y., Cao, X.: Optimal p-ary cyclic codes with two zeros. Appl. Algrbr. Eng. Comm. 34, 129–138 (2023)
Macwilliams, F., Sloane, N.: The theory of error-correcting codes. Elsevier, North Holland (1977)
Ireland, K., Rosen, M.: A classical introduction to modern number theory. Springer-Verlag, Germany (1990)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, New York, USA (1977)
Li, L., Zhu, S., Liu, L.: Three classes of optimal ternary cyclic codes and the weight distributions of their duals. Chinese J. Electron. 28(4), 674–681 (2019)
Delsarte, P.: On subfield subcodes of modified Reed-Solomon code. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)
Yang, S., Yao, Z., Zhao, C.: The weight distributions of two classes of p-ary cyclic codes with few weights. Finite Fields Appl. 44, 76–91 (2017)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. U21A20428, No. 12171134, No. 62201009 and No. 12201170) and the Natural Science Foundation of Anhui Province (No. 2108085QA06, No. 2108085QA03).
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Wu, T., Liu, L. & Li, L. Several classes of optimal cyclic codes with three zeros. AAECC (2023). https://doi.org/10.1007/s00200-023-00636-0
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DOI: https://doi.org/10.1007/s00200-023-00636-0