Optimal constacyclic codes with minimum distance four
Tóm tắt
Let
$$n=2(p^m-1)/(p-1)$$
, where p is an odd prime and
$$m>1$$
is a positive integer. In this paper, we research optimal p-ary constacyclic codes with two zeros. Two classes of optimal p-ary
$$[n,n-2m,4]$$
constacyclic codes are presented by searching the solutions of certain congruence equations over
$$\mathbb {F}_{p^m}$$
. Four explicit constructions of optimal constacyclic codes with such parameters are provided. The dual codes of a subclass of these constacyclic codes are also investigated.
Từ khóa
Tài liệu tham khảo
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)
Delsarte, P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)
Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013)
Fan, C., Li, N., Zhou, Z.: A class of optimal ternary cyclic codes and their duals. Finite Fields Appl. 37, 193–202 (2016)
Fang, W., Wen, J., Fu, F.: A \(q\) -polynomial approach to constacyclic codes. Finite Fields Appl. 47, 161–182 (2017)
Han, D., Yan, H.: On an open problem about a class of optimal ternary cyclic codes. Finite Fields Appl. 59, 335–343 (2019)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge Univ. Press, Cambridge (2003)
Li, C., Ding, C., Li, S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017)
Li, C., Li, N., Helleseth, T., Ding, C.: The weight distributions of several classes of cyclic codes from APN monomials. IEEE Trans. Inf. Theory 60(8), 4710–4721 (2014)
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: Signal Design and its Applications in Communications (IWSDA), 2015 Seventh Intertional Workshop on Signal (2015)
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 Commun. Lett. 24(7), 1387–1390 (2020)
Liu, Y., Cao, X., Lu, W.: On some conjectures about optimal ternary cyclic codes. Des. Codes Cryptogr. 88(2), 297–309 (2020)
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
Luo, J., Feng, K.: Cyclic codes and sequence from generalized Coulter–Matthews function. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008)
Wang, L., Wu, G.: Several classes of optimal ternary cyclic codes with minimum distance four. Finite Fields Appl. 40, 126–137 (2016)
Xu, G., Cao, X., Xu, S.: Optimal \(p\)-ary cyclic codes with minimum distance four from monomials. Cryptogr. Commun. 8(4), 541–554 (2016)
Yan, H., Han, D.: New ternary power mapping with differential uniformity \(\Delta _f \le \) 3 and related optimal cyclic codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 102(6), 849–853 (2019)
Yan, H., Zhou, Z., Du, X.: A family of optimal ternary cyclic codes from the NIHO-type exponent. Finite Fields Appl. 54, 101–112 (2018)
Zha, Z., Hu, L.: New classes of optimal ternary cyclic codes with minimum distance four. Finite Fields Appl. 64, 101671 (2020)
Zha, Z., Hu, L., Liu, Y., Cao, X.: Further results on optimal cyclic codes. Finite Fields Appl. 75, 101898 (2021)
Zhao, H., Luo, R., Sun, T.: Two families of optimal ternary cyclic codes with minimal distance four. Finite Fields Appl. 79, 101995 (2022)
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)
Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)