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Several classes of optimal ternary cyclic codes with minimal distance four. (English) Zbl 1405.94126

Summary: In this paper, by analyzing the solutions of certain equations over \(\mathbb{F}_{3^m}\), we present four classes of optimal ternary cyclic codes with parameters \([3^m-1, 3^m-1-2m, 4]\). It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results.

MSC:

94B15 Cyclic codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B75 Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) to coding theory
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[1] 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) · Zbl 1192.94114
[2] Chien, R. T., Cyclic decoding procedure for the Bose-Chaudhuri-Hocquenghem codes, IEEE Trans. Inf. Theory, 10, 4, 357-363 (1964) · Zbl 0125.09503
[3] Ding, C.; Gao, Y.; Zhou, Z., Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59, 12, 7940-7946 (2013) · Zbl 1364.94651
[4] Ding, C.; Helleseth, T., Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59, 9, 5898-5904 (2013) · Zbl 1364.94652
[5] Ding, C.; Ling, S., A q-polynomial approach to cyclic codes, Finite Fields Appl., 20, 3, 1-14 (2013) · Zbl 1308.94107
[6] Feng, T., On cyclic codes of length \(2^{2^r} - 1\) with two zeros whose dual codes have three weights, Des. Codes Cryptogr., 62, 3, 253-258 (2012) · Zbl 1282.94096
[7] Forney, G. D., On decoding BCH codes, IEEE Trans. Inf. Theory, 11, 4, 549-557 (1965) · Zbl 0143.41401
[8] Huffman, W. C.; Pless, V., Fundamentals of Error-Correcting Codes (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, U.K. · Zbl 1099.94030
[9] Jia, Y.; Ling, S.; Xing, C., On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57, 4, 2243-2251 (2011) · Zbl 1366.94639
[10] 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) · Zbl 1360.94402
[11] Li, N.; Li, C.; Helleseth, T.; Ding, C.; Tang, X. H., Optimal ternary cyclic codes with minimum distance four and five, Finite Fields Appl., 30, 100-120 (2014) · Zbl 1354.94067
[12] Prange, E., Some cyclic error-correcting codes with simple decoding algorithms (1958), Air Force Cambridge Research Center-TN-58-156: Air Force Cambridge Research Center-TN-58-156 Cambridge, MA, USA
[13] Zeng, X.; Shan, J.; Hu, L., A triple-error-correcting cyclic code from the Gold and Kasami-Welch APN power functions, Finite Fields Appl., 18, 1, 70-92 (2012) · Zbl 1246.94053
[14] Zhou, Z.; Ding, C., A class of three-weight cyclic codes, Finite Fields Appl., 25, 79-93 (2014) · Zbl 1305.94112
[15] Zhou, Z.; Ding, C., Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61, 10, 4120-4126 (2013)
[16] Zhou, Z.; Ding, C.; Luo, J.; Zhang, A., A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59, 10, 6674-6682 (2013) · Zbl 1364.94666
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