Two families of optimal ternary cyclic codes with minimal distance four. (English) Zbl 1485.94168
Summary: In this paper, the two families of optimal ternary cyclic codes with parameters of \([3^m -1, 3^m -1-2m, 4]\) are presented. Some conclusions proposed by L. Wang and G. Wu in 2016 [Finite Fields Appl. 40, 126–137 (2016; Zbl 1405.94126)] are analyzed and refined, and the remaining problem in it is solved. In addition, a conjecture proposed by C. Ding and T. Helleseth in 2013 [IEEE Trans. Inf. Theory 59, No. 9, 5898–5904 (2013; Zbl 1364.94652)] about a class of optimal ternary cyclic codes \(\mathcal{C}_{(1,e)}\) in which \(e = (3^{m-1} -1) / 2 + 3^h +1\) with parameters \([3^m -1, 3^m -2m-1, 4]\) is analyzed, where \(0 \leq h \leq m-1\).
MSC:
| 94B15 | Cyclic codes |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 12E12 | Equations in general fields |
References:
| [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] | Huffman, W. C.; Pless, V., Fundamentals of Error-Correcting Codes (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, U.K. · Zbl 1099.94030 |
| [4] | Ding, C.; Yang, J., Hamming weights in irreducible cyclic codes, Discrete Math., 313, 4, 434-446 (2013) · Zbl 1269.94040 |
| [5] | Forney, G. D., On decoding BCH codes, IEEE Trans. Inf. Theory, 11, 4, 549-557 (1965) · Zbl 0143.41401 |
| [6] | Prange, E., Some cyclic error-correcting codes with simple decoding algorithms, (Air Force Cambridge Research Center-TN-58-156. Air Force Cambridge Research Center-TN-58-156, Cambridge, MA, USA (1958)) |
| [7] | 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 |
| [8] | 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 |
| [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] | Ding, C.; Ling, S., A q-polynomial approach to cyclic codes, Finite Fields Appl., 20, 3, 1-14 (2013) · Zbl 1308.94107 |
| [11] | Blondeau, C.; Canteaut, A.; Charpin, P., Differential properties of power functions, (Proceedings of the 2010 IEEE International Symposium on Information Theory. Proceedings of the 2010 IEEE International Symposium on Information Theory, ISIT 10, Austin, USA (June 2010)), 2478-2482 |
| [12] | Berger, T. P.; Canteaut, A.; Charpin, P.; Laigle-Chapuy, Y., On almost perfect nonlinear functions, IEEE Trans. Inf. Theory, 52, 9, 4160-4170 (2006) · Zbl 1184.94224 |
| [13] | Ding, C.; Helleseth, T., Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59, 9, 5898-5904 (2013) · Zbl 1364.94652 |
| [14] | Helleseth, T.; Rong, C.; Sandberg, D., New families of almost perfect nonlinear power mappings, IEEE Trans. Inf. Theory, 4, 2, 475-485 (1999) · Zbl 0960.11051 |
| [15] | Zhou, Z.; Ding, C., A class of three-weight cyclic codes, Finite Fields Appl., 25, 79-93 (2014) · Zbl 1305.94112 |
| [16] | Zhou, Z.; Ding, C., Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61, 10, 4120-4126 (2013) |
| [17] | 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 |
| [18] | Li, N.; Zhou, Z.; Helleseth, T., On a conjecture about a class of optimal ternary cyclic codes, (Seventh International Workshop on Signal Design and Its Applications in Communications (2015)), 62-65 |
| [19] | Wang, L.; Wu, G., Several classes of optimal ternary cyclic codes with minimal distance four, Finite Fields Appl., 40, 126-137 (2016) · Zbl 1405.94126 |
| [20] | Han, D.; Yan, H., On a open problem about a class of optimal ternary cyclic codes, Finite Fields Appl., 59, 335-343 (2019) · Zbl 1421.94102 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
