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The weight distributions of constacyclic 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. | State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China |
4. | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China |
5. | Science and Technology on Information Assurance Laboratory, Beijing 100072, China |
Let $\Bbb F_q$ be a finite field with $q$ elements. Suppose that $a, λ∈ \Bbb F_q^*$, $a^n=λ$ with $n|(q-1)$. In this paper, we determine the weight distribution of a class of $λ$-constacyclic codes of length $nm$ with the parity check polynomial $h(x)=(x^m-aξ^{st})(x^m-aξ^{s(t+1)})...(x^m-aξ^{s(t+r-1)})$ and $n>(r-1)m$, where $s,t, r$ are positive integers and $ξ∈ \Bbb F_q$ is a primitive n-th root of unity. Moreover, we give the weight distributions of $λ$-constacyclic codes of length $nm$ explicitly in several cases: (1) $r=1$, $n>1$; (2) $r=2$, $m=2$ and $n>2$; (3) $r=2$, $m=3$ and $n>3$; (4) $r=3$, $m=2$ and $n>4$.
References:
[1] |
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. |
[2] |
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. |
[3] |
B. Chen, H. Q. Dinh and H. Liu,
Repeated-root constacyclic codes of length $\ell p^s$ and their duals, Discr. Appl. Math., 177 (2014), 60-70.
doi: 10.1016/j.disc.2013.01.024. |
[4] |
B. Chen, H. Q. Dinh and H. Liu,
Repeated-root constacyclic codes of length $2\ell^mp^n$, Finite Fields Appl., 33 (2015), 137-159.
doi: 10.1016/j.ffa.2014.11.006. |
[5] |
B. Chen, Y. Fan, L. Lin and H. Liu,
Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.
|
[6] |
C. Ding,
The weight distributions of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.
doi: 10.1109/TIT.2008.2011511. |
[7] |
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. |
[8] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[9] |
H. Q. Dinh,
On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14 (2008), 22-40.
doi: 10.1016/j.ffa.2007.07.001. |
[10] |
H. Q. Dinh,
Repeated-root constacyclic codes of length $2p^s$, Finite Fields Appl., 18 (2012), 133-143.
doi: 10.1016/j.ffa.2011.07.003. |
[11] |
H. Q. Dinh,
Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discr. Math., 313 (2013), 983-991.
doi: 10.1016/j.disc.2013.01.024. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
C. Li, X. Zeng and L. Hu,
A class of binary cyclic codes with five weights}, Sci. China Math., 53 (2010), 3279-3286.
doi: 10.1007/s11425-010-4062-z. |
[16] |
R. Lidl and H. Niederreiter,
Finite Fields Cambridge Univ. Press, Cambridge, 2008. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
X. Wang, D. Zheng, L. Hu and X. 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. |
[24] |
J. Wolfmann,
Weight distributions of some binary primitive cyclic codes, IEEE Trans. Inf. Theory, 40 (2004), 2068-2071.
doi: 10.1109/18.340482. |
[25] |
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. |
[26] |
J. Yang, M. Xiong and C. Ding,
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. |
[27] |
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. |
[28] |
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.2012.06.001. |
[29] |
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. |
[30] |
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. |
[31] |
D. Zheng, X. Wang, L. Hu and X. Zeng,
The weight distributions of two classes of p-ary cyclic codes, Finite Fields Appl., 29 (2014), 202-224.
doi: 10.1016/j.ffa.2014.05.001. |
[32] |
D. Zheng, X. Wang, X. Zeng and L. Hu,
The weight distributions of a family of p-ary cyclic codes, Des. Codes Crypt., 75 (2015), 263-275.
doi: 10.1007/s10623-013-9908-2. |
[33] |
X. Zhu, Q. Yue and L. Hu,
Weight distribution of cyclic codes of length $tl^m$, Discr. Math., 338 (2015), 844-856.
|
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
B. Chen, H. Q. Dinh and H. Liu,
Repeated-root constacyclic codes of length $\ell p^s$ and their duals, Discr. Appl. Math., 177 (2014), 60-70.
doi: 10.1016/j.disc.2013.01.024. |
[4] |
B. Chen, H. Q. Dinh and H. Liu,
Repeated-root constacyclic codes of length $2\ell^mp^n$, Finite Fields Appl., 33 (2015), 137-159.
doi: 10.1016/j.ffa.2014.11.006. |
[5] |
B. Chen, Y. Fan, L. Lin and H. Liu,
Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.
|
[6] |
C. Ding,
The weight distributions of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.
doi: 10.1109/TIT.2008.2011511. |
[7] |
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. |
[8] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discr. Math., 313 (2013), 434-446.
doi: 10.1016/j.disc.2012.11.009. |
[9] |
H. Q. Dinh,
On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14 (2008), 22-40.
doi: 10.1016/j.ffa.2007.07.001. |
[10] |
H. Q. Dinh,
Repeated-root constacyclic codes of length $2p^s$, Finite Fields Appl., 18 (2012), 133-143.
doi: 10.1016/j.ffa.2011.07.003. |
[11] |
H. Q. Dinh,
Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discr. Math., 313 (2013), 983-991.
doi: 10.1016/j.disc.2013.01.024. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
C. Li, X. Zeng and L. Hu,
A class of binary cyclic codes with five weights}, Sci. China Math., 53 (2010), 3279-3286.
doi: 10.1007/s11425-010-4062-z. |
[16] |
R. Lidl and H. Niederreiter,
Finite Fields Cambridge Univ. Press, Cambridge, 2008. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
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. |
[23] |
X. Wang, D. Zheng, L. Hu and X. 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. |
[24] |
J. Wolfmann,
Weight distributions of some binary primitive cyclic codes, IEEE Trans. Inf. Theory, 40 (2004), 2068-2071.
doi: 10.1109/18.340482. |
[25] |
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. |
[26] |
J. Yang, M. Xiong and C. Ding,
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. |
[27] |
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. |
[28] |
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.2012.06.001. |
[29] |
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. |
[30] |
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. |
[31] |
D. Zheng, X. Wang, L. Hu and X. Zeng,
The weight distributions of two classes of p-ary cyclic codes, Finite Fields Appl., 29 (2014), 202-224.
doi: 10.1016/j.ffa.2014.05.001. |
[32] |
D. Zheng, X. Wang, X. Zeng and L. Hu,
The weight distributions of a family of p-ary cyclic codes, Des. Codes Crypt., 75 (2015), 263-275.
doi: 10.1007/s10623-013-9908-2. |
[33] |
X. Zhu, Q. Yue and L. Hu,
Weight distribution of cyclic codes of length $tl^m$, Discr. Math., 338 (2015), 844-856.
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Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
Weight | Frequency |
0 | 1 |
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