# American Institute of Mathematical Sciences

August  2014, 8(3): 297-312. doi: 10.3934/amc.2014.8.297

## Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)

 1 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China, China 2 Science and Technology on Information Assurance Laboratory, Beijing, 100072, China

Received  April 2013 Revised  December 2013 Published  August 2014

In this paper, we always assume that $p=6f+1$ is a prime. First, we calculate the values of exponential sums of cyclotomic classes of orders 3 and 6 over an extension field of GF(3). Then, we give a formula to compute the linear complexity of all $p^{n+1}$-periodic generalized cyclotomic sequences of order 6 over GF(3). After that, we compute the linear complexity and the minimal polynomial of a $p^{n+1}$-periodic, balanced and generalized cyclotomic sequence of order 6 over GF(3), which is analogous to a generalized Sidelnikov's sequence. At last, we give some BCH codes with prime length $p$ from cyclotomic sequences of orders three and six.
Citation: Liqin Hu, Qin Yue, Fengmei Liu. Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3). Advances in Mathematics of Communications, 2014, 8 (3) : 297-312. doi: 10.3934/amc.2014.8.297
##### References:
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##### References:
 [1] E. Bai, X. Liu and G. Xiao, Linear complexity of new generalized cyclotomic sequences of order two of length pq, IEEE Trans. Inform. Theory, 51 (2005), 1849-1853. doi: 10.1109/TIT.2005.846450. [2] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, J. Wiley and Sons Company, New York, 1997. [3] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order $2$, Finite Fields Appl., 3 (1997), 159-174. doi: 10.1006/ffta.1997.0181. [4] C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inform. Theory, 44 (1998), 1698-1702. doi: 10.1109/18.681354. [5] C. Ding, Pattern distribution of Legendre sequences, IEEE Trans. Inform. Theory, 44 (1998), 1693-1698. doi: 10.1109/18.681353. [6] C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960. doi: 10.1109/TIT.2008.2011511. [7] C. Ding, Cyclic codes from cyclotomic sequences of order four, Finite Fields Appl., 23 (2013), 8-34. doi: 10.1016/j.ffa.2013.03.006. [8] C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166. doi: 10.1006/ffta.1998.0207. [9] C. Ding, T. Helleseth and K. Y. Lam, Several classes of sequences with three-level autocorrelation, IEEE Trans. Inform. Theory, 45 (1999), 2606-2612. doi: 10.1109/18.796414. [10] C. Ding, T. Helleseth and H. M. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory, 47 (2001), 428-433. doi: 10.1109/18.904555. [11] L. E. Dickson, Cyclotomy, higher congruences and Waring's problem, Amer. J. math., 57 (1935), 391-424. doi: 10.2307/2371217. [12] V. A. Edemskii, On the linear complexity of binary sequences on the basis of biquadratic and sextic residue classes, Discrete Math. Appl., 20 (2010), 75-84. doi: 10.1515/DMA.2010.004. [13] V. A. Edemskii, About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$, Des. Codes Cryptogr., 61 (2011), 251-260. doi: 10.1007/s10623-010-9474-9. [14] K. Feng and F. Liu, Algebra and Communication, Higher education press, Beijing, 2005. [15] M. Hall, Combinatorial Theory, Blaisdell Company, Waltham, 1967. [16] K. Ireland and M. Rosen, A classical introduction to modern number theory, Second edition, Springer-Verlag, 2003. doi: 10.1007/978-1-4757-2103-4. [17] J. H. Kim and H. Y. Song, On the linear complexity of Hall's sextic residue sequences, IEEE Trans. Inform. Theory, 47 (2001), 2094-2096. doi: 10.1109/18.930950. [18] A. Lempel, M. Cohn and W. L. Estman, A class of binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory, 23 (1997), 38-42. doi: 10.1109/TIT.1977.1055672. [19] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Company, 1983. [20] F. Liu, D. Y. Peng, X. H. Tang and X. H. Niu, On the autocorrelation and the linear complexity of $q$-ary prime $n$-square sequences, in Sequences and their Applications- SETA 2010, 2010, 139-150, doi: 10.1007/978-3-642-15874-2_11. [21] V. M. Sidelnikov, Some k-valued pseudo-random sequences and nearly equidistance codes, Probl. Inform. Transm., 5 (1969), 12-16. [22] A. L. Whiteman, The cyclotomic numbers of order twelve, Acta Arith., 6 (1960), 53-67.
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