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The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes
1. | School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, Shandong, 277160, China |
2. | State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China |
3. | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China |
$ l $ |
$ l\equiv 3\pmod 4 $ |
$ l\ne 3 $ |
$ N = l^m $ |
$ m $ |
$ f = \phi(N)/2 $ |
$ p $ |
$ N $ |
$ q = p^f $ |
$ \phi(\cdot) $ |
$ \alpha $ |
$ \Bbb F_{q} $ |
$ C_0^{(N,q)} = \langle \alpha^N\rangle $ |
$ \Bbb F_q^* $ |
$ C_i^{(N,q)} = \alpha^i\langle \alpha^N\rangle $ |
$ i = 0,\ldots, N-1 $ |
$ \eta_i^{(N, q)} = \sum_{x \in C_i^{(N,q)}}\psi(x) $ |
$ i = 0,1,\cdots, N-1 $ |
$ \psi $ |
$ \Bbb F_{q} $ |
$ \eta_i^{(2N, q)} $ |
$ i = 0,1,\cdots, 2N-1 $ |
$ q $ |
$ p $ |
$ p $ |
$ \mathcal{C}_{D} = \{C = ( \operatorname{Tr}_{q/p}(c x_1), \operatorname{Tr}_{q/p}(cx_2),\ldots, \operatorname{Tr}_{q/p}(cx_n)):c\in \Bbb{F}_{q}\}, $ |
$ D $ |
$ D = \{x\in \Bbb{F}_{q}^{*}: \operatorname{Tr}_{q/p}(x^{\frac{q-1}{l^{m}}}) = 0\} $ |
$ \operatorname{Tr}_{q/p} $ |
$ \Bbb F_{q} $ |
$ \Bbb F_p $ |
References:
[1] |
L. Baumert and J. Mykkeltveit,
Weight distributions of some irreducible cyclic codes, DSN Progr. Rep., 16 (1973), 128-131.
|
[2] |
B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997. |
[3] |
H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02945-9. |
[4] |
C. S. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[5] |
C. S. Ding, Y. Liu, C. L. Ma and L. W. Zeng,
The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.
doi: 10.1109/TIT.2011.2165314. |
[6] |
C. S. Ding, C. L. Li, N. Li and Z. C. Zhou,
Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.
doi: 10.1016/j.disc.2015.09.001. |
[7] |
T. Feng and Q. Xiang,
Strongly regular graphs from unions of cyclotomic classes, Journal of Combinatorial Theory Series B, 102 (2012), 982-995.
doi: 10.1016/j.jctb.2011.10.006. |
[8] |
Z. L. Heng and Q. Yue,
A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.
doi: 10.1109/LCOMM.2015.2455032. |
[9] |
Z. L. Heng and Q. Yue,
Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.
doi: 10.1016/j.ffa.2015.12.002. |
[10] |
Z. L. Heng and Q. Yue,
Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.
doi: 10.1007/s10623-016-0222-7. |
[11] |
L. Q. Hu, Q. Yue and M. H. Wang,
The linear complexity of Whiteman's generalize cyclotomic sequences of period $p^{m+1}q^{n+1}$, IEEE Trans. Inform. Theory, 58 (2012), 5534-5543.
doi: 10.1109/TIT.2012.2196254. |
[12] |
P. Langevin,
Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64.
doi: 10.1006/jnth.1997.2078. |
[13] |
C. J. Li and Q. Yue,
The Walsh transform of a class of monomial functions and cyclic codes, Cryptogr. Commun., 7 (2015), 217-228.
doi: 10.1007/s12095-014-0109-2. |
[14] |
C. J. Li, Q. Yue and F. W. 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. |
[15] |
F. W. Li, Q. Yue and F. M. Liu,
The weight distribution of a class of cyclic codes containing a subclass with optimal parameters, Finite Fields Appl., 45 (2017), 183-202.
doi: 10.1016/j.ffa.2016.12.004. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. |
[17] |
Y. W. Liu and Z. H. Liu,
On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.
doi: 10.3934/amc.2018025. |
[18] |
J. Q. Luo and K. Q. 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. |
[19] |
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. |
[20] |
G. Myerson,
Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.
doi: 10.4064/aa-39-3-251-264. |
[21] |
T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967. |
[22] |
Q. Y. Wang, K. L. Ding, D. D. Lin and R. Xue,
A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322.
doi: 10.1007/s12095-015-0180-3. |
[23] |
Q. Y. Wang, K. L. Ding and R. Xue,
Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100.
doi: 10.1109/LCOMM.2015.2431253. |
[24] |
X. Q. Wang, D. B. Zheng, L. Hu and X. Y. Zeng,
The weight distributions of two classes of binary cyclic codes, Finite Fields Appl., 34 (2015), 192-207.
doi: 10.1016/j.ffa.2015.01.012. |
[25] |
M. S. 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. |
[26] |
J. Yang and L. L. Xia,
Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.
doi: 10.1007/s11425-010-3155-z. |
[27] |
J. Yang, M. S. Xiong, C. S. Ding and J. Q. Luo,
Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.
doi: 10.1109/TIT.2013.2266731. |
[28] |
S. D. Yang, X. L. Kong and C. M. Tang,
A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[29] |
S. D. Yang and Z.-A. Yao,
Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[30] |
Z. C. Zhou and C. S. Ding,
A class of three-weight cyclic codes, Finite Fields Appli., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
[31] |
Z. C. Zhou, A. X. Zhang and C. S. Ding,
The weight enumerator of three families of cyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 6002-6009.
doi: 10.1109/TIT.2013.2262095. |
show all references
References:
[1] |
L. Baumert and J. Mykkeltveit,
Weight distributions of some irreducible cyclic codes, DSN Progr. Rep., 16 (1973), 128-131.
|
[2] |
B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, New York, John Wiley & Sons Company, 1997. |
[3] |
H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02945-9. |
[4] |
C. S. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[5] |
C. S. Ding, Y. Liu, C. L. Ma and L. W. Zeng,
The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.
doi: 10.1109/TIT.2011.2165314. |
[6] |
C. S. Ding, C. L. Li, N. Li and Z. C. Zhou,
Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.
doi: 10.1016/j.disc.2015.09.001. |
[7] |
T. Feng and Q. Xiang,
Strongly regular graphs from unions of cyclotomic classes, Journal of Combinatorial Theory Series B, 102 (2012), 982-995.
doi: 10.1016/j.jctb.2011.10.006. |
[8] |
Z. L. Heng and Q. Yue,
A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491.
doi: 10.1109/LCOMM.2015.2455032. |
[9] |
Z. L. Heng and Q. Yue,
Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.
doi: 10.1016/j.ffa.2015.12.002. |
[10] |
Z. L. Heng and Q. Yue,
Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307-326.
doi: 10.1007/s10623-016-0222-7. |
[11] |
L. Q. Hu, Q. Yue and M. H. Wang,
The linear complexity of Whiteman's generalize cyclotomic sequences of period $p^{m+1}q^{n+1}$, IEEE Trans. Inform. Theory, 58 (2012), 5534-5543.
doi: 10.1109/TIT.2012.2196254. |
[12] |
P. Langevin,
Caluls de certaines sommes de Gauss, J. Number theory, 63 (1997), 59-64.
doi: 10.1006/jnth.1997.2078. |
[13] |
C. J. Li and Q. Yue,
The Walsh transform of a class of monomial functions and cyclic codes, Cryptogr. Commun., 7 (2015), 217-228.
doi: 10.1007/s12095-014-0109-2. |
[14] |
C. J. Li, Q. Yue and F. W. 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. |
[15] |
F. W. Li, Q. Yue and F. M. Liu,
The weight distribution of a class of cyclic codes containing a subclass with optimal parameters, Finite Fields Appl., 45 (2017), 183-202.
doi: 10.1016/j.ffa.2016.12.004. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. |
[17] |
Y. W. Liu and Z. H. Liu,
On some classes of codes with a few weights, Adv. Math. Commun., 12 (2018), 415-428.
doi: 10.3934/amc.2018025. |
[18] |
J. Q. Luo and K. Q. 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. |
[19] |
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. |
[20] |
G. Myerson,
Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.
doi: 10.4064/aa-39-3-251-264. |
[21] |
T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967. |
[22] |
Q. Y. Wang, K. L. Ding, D. D. Lin and R. Xue,
A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322.
doi: 10.1007/s12095-015-0180-3. |
[23] |
Q. Y. Wang, K. L. Ding and R. Xue,
Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100.
doi: 10.1109/LCOMM.2015.2431253. |
[24] |
X. Q. Wang, D. B. Zheng, L. Hu and X. Y. Zeng,
The weight distributions of two classes of binary cyclic codes, Finite Fields Appl., 34 (2015), 192-207.
doi: 10.1016/j.ffa.2015.01.012. |
[25] |
M. S. 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. |
[26] |
J. Yang and L. L. Xia,
Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.
doi: 10.1007/s11425-010-3155-z. |
[27] |
J. Yang, M. S. Xiong, C. S. Ding and J. Q. Luo,
Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.
doi: 10.1109/TIT.2013.2266731. |
[28] |
S. D. Yang, X. L. Kong and C. M. Tang,
A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[29] |
S. D. Yang and Z.-A. Yao,
Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[30] |
Z. C. Zhou and C. S. Ding,
A class of three-weight cyclic codes, Finite Fields Appli., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
[31] |
Z. C. Zhou, A. X. Zhang and C. S. Ding,
The weight enumerator of three families of cyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 6002-6009.
doi: 10.1109/TIT.2013.2262095. |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
|
Weight | Frequency |
|
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