# American Institute of Mathematical Sciences

August  2015, 9(3): 341-352. doi: 10.3934/amc.2015.9.341

## Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates

 1 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China, China, China

Received  August 2014 Published  July 2015

Let $\Bbb F_{q^k}$ be a finite field and $\alpha$ a primitive element of $\Bbb F_{q^k}$, where $q=l^f$, $l$ is a prime power, and $f$ is a positive integer. Suppose that $N$ is a positive integer and $m_{g^{l^u}}(x)$ is the minimal polynomial of $g^{l^u}$ over $\Bbb F_q$ for $u=0, 1, \ldots, f-1$, where $g=\alpha^{-N}$. Let $\mathcal C$ be a cyclic code over $\Bbb F_q$ with check polynomial $$m_g(x)m_{g^l}(x) \cdots m_{g^{l^{f-1}}}(x).$$ In this paper, we shall present a method to determine the weight distribution of the cyclic code $\mathcal C$ in two cases: (1) $\gcd(\frac {q^k-1} {l-1}, N)=1$; (2) $l=2$ and $f=2$. Moreover, we will obtain a class of two-weight cyclic codes and a class of new three-weight cyclic codes.
Citation: Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341
##### References:
 [1] L. Baumert, W. Mills and R. Ward, Uniform cyclotomy, J. Number Theory, 14 (1982), 67-82. doi: 10.1016/0022-314X(82)90058-0. [2] N. Boston and G. McGuire, The weight distribution of cyclic codes with two zeros and zeta functions, J. Symb. Comput., 45 (2010), 723-733. doi: 10.1016/j.jsc.2010.03.007. [3] A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. [4] A. Canteaut, P. Charpin and H. Dobbertin, Weight divisibility of cyclic codes, highly nonlinear functions on $\mathbbF_{2^m}$ and crosscorrelation of maximum-length sequences, SIAM J. Discrete Math., 13 (2000), 105-138. doi: 10.1137/S0895480198350057. [5] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130. [6] 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. [7] P. Charpin, Cyclic codes with few weights and Niho exponents, J. Combin. Theory Ser. A, 108 (2004), 247-259. doi: 10.1016/j.jcta.2004.07.001. [8] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576. [9] C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. [10] C. Ding, Y. Liu, C. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006. doi: 10.1109/TIT.2011.2165314. [11] C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446. doi: 10.1016/j.disc.2012.11.009. [12] C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612. doi: 10.1109/TIT.2010.2048504. [13] 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. [14] T. Feng, On cyclic codes of length $2^{2^r}-1$ with two zeros whose dual codes have three weights, Des. Codes Cryptogr., 62 (2012), 253-258. doi: 10.1007/s10623-011-9514-0. [15] T. Feng and K. Momihara, Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums, IEEE Trans. Inf. Theory, 59 (2013), 5980-5984. doi: 10.1109/TIT.2013.2259538. [16] É. Fouvry and J. Klüners, On the 4-rank of class groups of quadratic number fields, Invent. Math., 167 (2007), 455-513. doi: 10.1007/s00222-006-0021-2. [17] C. Li, N. Li, T. Helleseth and C. Ding, 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. [18] C. Li and Q. Yue, Weight distribution of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303. doi: 10.1109/TIT.2013.2287211. [19] C. Li, Q. Yue and F. Li, Hamming weights of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 60 (2014), 3895-3902. doi: 10.1109/TIT.2014.2317785. [20] C. Li, Q. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114. doi: 10.1016/j.ffa.2014.01.009. [21] S. Li, S. Hu, T. Feng and G. Ge, The weight distribution of a class of cyclic codes related to Hermitian forms graphs, IEEE Trans. Inf. Theory, 59 (2013), 3064-3067. doi: 10.1109/TIT.2013.2242957. [22] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983. [23] 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. [24] J. Luo, Y. Tang and H. Wang, Cyclic codes and sequences: the generalized Kasami case, IEEE Trans. Inf. Theory, 56 (2010), 2130-2142. doi: 10.1109/TIT.2010.2043783. [25] 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. [26] G. McGuire, On three weights in cyclic codes with two zeros, Finite Fields Appl., 10 (2004), 97-104. doi: 10.1016/S1071-5797(03)00045-5. [27] G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264. [28] G. Vega, Two-weight classes cyclic codes constructed as the direct sum of two one-weight cyclic codes, Finite Fields Appl., 14 (2008), 785-797. doi: 10.1016/j.ffa.2008.01.002. [29] G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869. doi: 10.1109/TIT.2012.2193376. [30] G. Vega and J. Wolfmann, New classes of 2-weight cyclic codes, Des. Codes Cryptogr., 42 (2007), 327-344. doi: 10.1007/s10623-007-9038-9. [31] 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. [32] L. Xia and J. Yang, Cyclotomic problem, Gauss sums and Legendre curve, Sci. China Math., 56 (2013), 1485-1508. doi: 10.1007/s11425-013-4653-6. [33] M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945. doi: 10.1016/j.ffa.2012.06.001. [34] M. Xiong, The weight distributions of a class of cyclic codes II, Des. Codes Cryptogr., 72 (2014), 511-528. doi: 10.1007/s10623-012-9785-0. [35] M. Xiong, The weight distributions of a class of cyclic codes III, Finite Fields Appl., 21 (2013), 84-96. doi: 10.1016/j.ffa.2013.01.004. [36] J. Yang, M. Xiong, C. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993. doi: 10.1109/TIT.2013.2266731. [37] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 52 (2006), 712-717. doi: 10.1109/TIT.2005.862125. [38] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212. doi: 10.1109/TIT.2005.860412. [39] X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73. doi: 10.1016/j.ffa.2009.12.001. [40] 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. [41] 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] L. Baumert, W. Mills and R. Ward, Uniform cyclotomy, J. Number Theory, 14 (1982), 67-82. doi: 10.1016/0022-314X(82)90058-0. [2] N. Boston and G. McGuire, The weight distribution of cyclic codes with two zeros and zeta functions, J. Symb. Comput., 45 (2010), 723-733. doi: 10.1016/j.jsc.2010.03.007. [3] A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. [4] A. Canteaut, P. Charpin and H. Dobbertin, Weight divisibility of cyclic codes, highly nonlinear functions on $\mathbbF_{2^m}$ and crosscorrelation of maximum-length sequences, SIAM J. Discrete Math., 13 (2000), 105-138. doi: 10.1137/S0895480198350057. [5] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130. [6] 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. [7] P. Charpin, Cyclic codes with few weights and Niho exponents, J. Combin. Theory Ser. A, 108 (2004), 247-259. doi: 10.1016/j.jcta.2004.07.001. [8] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576. [9] C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. [10] C. Ding, Y. Liu, C. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000-8006. doi: 10.1109/TIT.2011.2165314. [11] C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446. doi: 10.1016/j.disc.2012.11.009. [12] C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612. doi: 10.1109/TIT.2010.2048504. [13] 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. [14] T. Feng, On cyclic codes of length $2^{2^r}-1$ with two zeros whose dual codes have three weights, Des. Codes Cryptogr., 62 (2012), 253-258. doi: 10.1007/s10623-011-9514-0. [15] T. Feng and K. Momihara, Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums, IEEE Trans. Inf. Theory, 59 (2013), 5980-5984. doi: 10.1109/TIT.2013.2259538. [16] É. Fouvry and J. Klüners, On the 4-rank of class groups of quadratic number fields, Invent. Math., 167 (2007), 455-513. doi: 10.1007/s00222-006-0021-2. [17] C. Li, N. Li, T. Helleseth and C. Ding, 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. [18] C. Li and Q. Yue, Weight distribution of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303. doi: 10.1109/TIT.2013.2287211. [19] C. Li, Q. Yue and F. Li, Hamming weights of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 60 (2014), 3895-3902. doi: 10.1109/TIT.2014.2317785. [20] C. Li, Q. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114. doi: 10.1016/j.ffa.2014.01.009. [21] S. Li, S. Hu, T. Feng and G. Ge, The weight distribution of a class of cyclic codes related to Hermitian forms graphs, IEEE Trans. Inf. Theory, 59 (2013), 3064-3067. doi: 10.1109/TIT.2013.2242957. [22] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc., 1983. [23] 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. [24] J. Luo, Y. Tang and H. Wang, Cyclic codes and sequences: the generalized Kasami case, IEEE Trans. Inf. Theory, 56 (2010), 2130-2142. doi: 10.1109/TIT.2010.2043783. [25] 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. [26] G. McGuire, On three weights in cyclic codes with two zeros, Finite Fields Appl., 10 (2004), 97-104. doi: 10.1016/S1071-5797(03)00045-5. [27] G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264. [28] G. Vega, Two-weight classes cyclic codes constructed as the direct sum of two one-weight cyclic codes, Finite Fields Appl., 14 (2008), 785-797. doi: 10.1016/j.ffa.2008.01.002. [29] G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inf. Theory, 58 (2012), 4862-4869. doi: 10.1109/TIT.2012.2193376. [30] G. Vega and J. Wolfmann, New classes of 2-weight cyclic codes, Des. Codes Cryptogr., 42 (2007), 327-344. doi: 10.1007/s10623-007-9038-9. [31] 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. [32] L. Xia and J. Yang, Cyclotomic problem, Gauss sums and Legendre curve, Sci. China Math., 56 (2013), 1485-1508. doi: 10.1007/s11425-013-4653-6. [33] M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945. doi: 10.1016/j.ffa.2012.06.001. [34] M. Xiong, The weight distributions of a class of cyclic codes II, Des. Codes Cryptogr., 72 (2014), 511-528. doi: 10.1007/s10623-012-9785-0. [35] M. Xiong, The weight distributions of a class of cyclic codes III, Finite Fields Appl., 21 (2013), 84-96. doi: 10.1016/j.ffa.2013.01.004. [36] J. Yang, M. Xiong, C. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inf. Theory, 59 (2013), 5985-5993. doi: 10.1109/TIT.2013.2266731. [37] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 52 (2006), 712-717. doi: 10.1109/TIT.2005.862125. [38] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212. doi: 10.1109/TIT.2005.860412. [39] X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73. doi: 10.1016/j.ffa.2009.12.001. [40] 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. [41] 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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