American Institute of Mathematical Sciences

February  2015, 9(1): 23-36. doi: 10.3934/amc.2015.9.23

Some new classes of cyclic codes with three or six weights

 1 Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China 2 Department of Informatics, University of Bergen, N-5020 Bergen, Norway 3 CIPSI, Department of Electrical Engineering and Computer Science, University of Stavanger, 4036 Stavanger, Norway

Received  December 2013 Revised  July 2014 Published  February 2015

In this paper, a class of three-weight cyclic codes over prime fields $\mathbb{F}_p$ of odd order whose duals have two zeros, and a class of six-weight cyclic codes whose duals have three zeros are presented. The weight distributions of these cyclic codes are derived.
Citation: Yongbo Xia, Tor Helleseth, Chunlei Li. Some new classes of cyclic codes with three or six weights. Advances in Mathematics of Communications, 2015, 9 (1) : 23-36. doi: 10.3934/amc.2015.9.23
References:
 [1] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722. [2] S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by $d = \frac{p^n+1}{p^k+1} + \frac{p^n-1}{2}$, IEICE Trans. Comm., B96 (2013), 2190-2197. [3] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576. [4] C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946. doi: 10.1109/TIT.2013.2281205. [5] K. Feng and J. Luo, Value distribution of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041. doi: 10.1109/TIT.2007.903153. [6] K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003. [7] Z. Hu, X. Li, D. Mills, E. N. Müller, W. Sun, W. Willems, Y. Yang and Z. Zhang, On the crosscorrelation of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. doi: 10.1007/s002000100073. [8] C. Li, N. Li, T. Helleseth and C. Ding, On the weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721. doi: 10.1109/TIT.2014.2329694. [9] R. Lidl and H. Niederreiter, Finite fields, in Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Amsterdam, 1983. [10] Y. Liu, H. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-014-9984-y. [11] J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424. [12] C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402. doi: 10.1109/TIT.2010.2090272. [13] E. N. Müller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295. doi: 10.1109/18.746820. [14] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179. [15] B. Wang, C. Tang, Y. Qi, Y. Yang and M. Xu, The weight distributions of cyclic codes and elliptic curves, IEEE Trans. Inf. Theory, 58 (2012), 7253-7259. doi: 10.1109/TIT.2012.2210386. [16] Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342. doi: 10.1007/s00200-010-0128-y. [17] Z. Zhou and C. Ding, Seven families of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126. [18] Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93. doi: 10.1016/j.ffa.2013.08.005. [19] Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682. doi: 10.1109/TIT.2013.2267722.

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References:
 [1] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722. [2] S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by $d = \frac{p^n+1}{p^k+1} + \frac{p^n-1}{2}$, IEICE Trans. Comm., B96 (2013), 2190-2197. [3] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576. [4] C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946. doi: 10.1109/TIT.2013.2281205. [5] K. Feng and J. Luo, Value distribution of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041. doi: 10.1109/TIT.2007.903153. [6] K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003. [7] Z. Hu, X. Li, D. Mills, E. N. Müller, W. Sun, W. Willems, Y. Yang and Z. Zhang, On the crosscorrelation of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. doi: 10.1007/s002000100073. [8] C. Li, N. Li, T. Helleseth and C. Ding, On the weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721. doi: 10.1109/TIT.2014.2329694. [9] R. Lidl and H. Niederreiter, Finite fields, in Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Amsterdam, 1983. [10] Y. Liu, H. Yan and C. Liu, A class of six-weight cyclic codes and their weight distribution,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-014-9984-y. [11] J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424. [12] C. Ma, L. Zeng, Y. Liu, D. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397-402. doi: 10.1109/TIT.2010.2090272. [13] E. N. Müller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295. doi: 10.1109/18.746820. [14] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179. [15] B. Wang, C. Tang, Y. Qi, Y. Yang and M. Xu, The weight distributions of cyclic codes and elliptic curves, IEEE Trans. Inf. Theory, 58 (2012), 7253-7259. doi: 10.1109/TIT.2012.2210386. [16] Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d = \frac{p^n+1}{p+1} - \frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342. doi: 10.1007/s00200-010-0128-y. [17] Z. Zhou and C. Ding, Seven families of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126. [18] Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93. doi: 10.1016/j.ffa.2013.08.005. [19] Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674-6682. doi: 10.1109/TIT.2013.2267722.
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