Abstract
In this paper, let p be a prime with \(p\ge 7\). We determine the weight distributions of cyclic codes of length 6 over \(\mathbb {F}_q\) and the minimum Hamming distances of all repeated-root cyclic codes of length \(6p^s\) over \(\mathbb {F}_q\), where q is a power of p and s is an integer. Furthermore, we find all maximum distance separable codes of length \(6p^s\).
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References
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)
Castagnoli, G., Massey, J.L., Schoeller, P.A., von Seemann, N.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory 37(2), 337–342 (1991)
Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finitefields. Finite Fields Appl. 18(6), 1217–1231 (2012)
Chen, B., Dinh, H.Q., Liu, H.: Repeated-root constacyclic codes of length \(lp^s\) and their duals. Discrete Appl. Math. 177, 60–70 (2014)
Chen, B., Dinh, H.Q., Liu, H.: Repeated-root constacyclic codes of length \(2l^mp^s\). Finite Fields Appl. 33, 137–159 (2015)
Chen, B., Lin, L., Liu, H.: Constacyclic symbol-pair codes: lower bounds and optimal constructions. IEEE Trans. Inf. Theory 62(12), 7661–7666 (2017)
Delsarte, P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)
Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005)
Ding, C., Yang, Y., Tang, X.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 56(7), 3605–3612 (2010)
Dinh, H.Q.: On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions. Finite Fields Appl. 14, 22–40 (2008)
Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). J. Algebra 324, 940–950 (2010)
Dinh, H.Q.: Repeated-root constacyclic codes of length \(2p^s\). Finite Fields Appl. 18, 940–950 (2012)
Dinh, H.Q.: Structure of repeated-root constacyclic codes of length \(3p^s\) and their duals. Discrete Math. 313, 983–991 (2013)
Dinh, H.Q.: Structure of repeated-root cyclic codes and negacyclic codes of length \(6p^s\) and their duals. Contemp. Math. 609, 69–87 (2014)
Dinh, H.Q., Wang, X., Liu, H., Sriboonchitta, S.: On the Hamming distances of repeated-root constacyclic codes of length \(4p^s\). Discrete Math. 342(5), 1456–1470 (2019)
Dinh, H.Q., Wang, X., Sirisrisakulchai, J.: On the Hamming distances of constacyclic codes of length \(5p^s\). IEEE Access (2020). https://doi.org/10.1109/ACCESS.2020.2976536
Kai, X., Zhu, S.: On the distance of cyclic codes of length \(2^e\) over \(Z_4\). Discrete Math. 310, 12–20 (2010)
Kløve, T.: Codes for Error Detection. World Scientific, Singapore (2007)
Li, C., Yue, Q., Li, F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014)
Liu, L., Li, L., Kai, X., Zhu, S.: Repeated-root constacyclic codes of length \(3lp^n\) and their dual codes. Finite Fields Appl. 42, 269–295 (2016)
Özadam, H., Özbudak, F.: The minimum Hamming distance of cyclic codes of length \(2p^s\). Appl. Algebra Eng. Commun. Comput. 5527, 92–100 (2009)
Raka, M.: A class of constacyclic codes over a finite field-II. Indian J. Pure Appl. Math. 46(6), 809–825 (2015)
Sharma, A.: Repeated-root constacyclic codes of length \(l^tp^s\) and their duals codes. Cryptogr. Commun. 7, 229–255 (2015)
Sharma, A., Rani, S.: On constacyclic codes over finite fields. Cryptogr. Commun. 8(4), 617–636 (2016)
Van Lint, J.H.: Repeated-root cyclic codes. IEEE Trans. Inf. Theory 37(2), 343–345 (1991)
Van Lint, J.H.: Introduction to Coding Theory. Springer, Berlin (2003)
Zhu, X., Yue, Q., Hu, L.: Weight distributions of cyclic codes of length \(tl^m\). Discrete Appl. Math. 338, 844–856 (2015)
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The paper was supported by National Natural Science Foundation of China (No. 61772015) and Foundation of Green wingceltis scholars of Zaozhuang University.
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Gao, Y., Yue, Q. & Li, F. The minimum Hamming distances of repeated-root cyclic codes of length \(6p^s\) and their MDS codes. J. Appl. Math. Comput. 65, 107–123 (2021). https://doi.org/10.1007/s12190-020-01383-y
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DOI: https://doi.org/10.1007/s12190-020-01383-y