American Institute of Mathematical Sciences

Advanced
August  2017, 11(3): 471-480. doi: 10.3934/amc.2017039

The weight distributions of constacyclic codes

Fengwei Li 1,2,3, , Qin Yue 2,3,4, and  Fengmei Liu 5,

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

Received  June 2015 Published  August 2017

Fund Project: The paper was supported by National Natural Science Foundation of China under Grants 11601475,61772015 and Foundation of Science and Technology on Information Assurance Laboratory under Grants KJ-15-009,6142112010202.

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 $n>4$.

Keywords: Finite field, Constacyclic code, weight distribution.
Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 11C08, 11C20.
Citation: Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
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. 

Table 1.  Weight distribution 
Weight Frequency
0 1
$n-1$ $2n(q-1)$
$n$ $2(q-1)(q+1-n)$
$2n-2$ $n^2(q-1)^2$
$2n-1$ $2n(q-1)^2(q+1-n)$
$2n$ $(q-1)^2(q+1-n)^2$
Table Options

Download as excel

Weight Frequency
0 1
$n-1$ $2n(q-1)$
$n$ $2(q-1)(q+1-n)$
$2n-2$ $n^2(q-1)^2$
$2n-1$ $2n(q-1)^2(q+1-n)$
$2n$ $(q-1)^2(q+1-n)^2$
Table 2.  Weight distribution 
Weight Frequency
0 1
$n-1$ $3n(q-1)$
$n$ $3(q-1)(q+1-n)$
$2n-2$ $3n^2(q-1)^2$
$2n-1$ $6n(q-1)^2(q+1-n)$
$2n$ $3(q-1)^2(q+1-n)^2$
$3n-3$ $n^3(q-1)^3$
$3n-2$ $3n^2(q-1)^3(q+1-n)$
$3n-1$ $3n(q-1)^3(q+1-n)^2$
$3n$ $(q-1)^3(q+1-n)^3$
Table Options

Download as excel

Weight Frequency
0 1
$n-1$ $3n(q-1)$
$n$ $3(q-1)(q+1-n)$
$2n-2$ $3n^2(q-1)^2$
$2n-1$ $6n(q-1)^2(q+1-n)$
$2n$ $3(q-1)^2(q+1-n)^2$
$3n-3$ $n^3(q-1)^3$
$3n-2$ $3n^2(q-1)^3(q+1-n)$
$3n-1$ $3n(q-1)^3(q+1-n)^2$
$3n$ $(q-1)^3(q+1-n)^3$
Table 3.  Weight distribution 
Weight Frequency
0 1
$n-2$ $n(n-1)(q-1)$
$n-1$ $2n(q-1)(q-n+2)$
$n$ $2(q-1)^3-2(n-3)(q-1)^2+(n-2)(n-3)(q-1)$
$2n-4$ $\frac {n^2(n-1)^2}4 (q-1)^2$
$2n-3$ $n^2(q-1)^2(n-1)(q-n+2)$
$2n-2$ $n^2(q-1)^2(q-n+2)^2+n(n-1)(q-1)^2[(q-1)^2-(n-3)(q-1)+C_{n-2}^2]$
$2n-1$ $2n(q-n+2)(q-1)^4-2n(n-3)(q-n+2)(q-1)^3$
$+n(n-2)(n-3)(q-n+2)(q-1)^2$
$2n$ $(q-1)^6+(n-3)^2(q-1)^4+\frac {(n-2)^2(n-3)^2}4 (q-1)^2+(n-2)(n-3)(q-1)^4$
$-2(n-3)(q-1)^5-(n-2)(n-3)^2(q-1)^3$
Table Options

Download as excel

Weight Frequency
0 1
$n-2$ $n(n-1)(q-1)$
$n-1$ $2n(q-1)(q-n+2)$
$n$ $2(q-1)^3-2(n-3)(q-1)^2+(n-2)(n-3)(q-1)$
$2n-4$ $\frac {n^2(n-1)^2}4 (q-1)^2$
$2n-3$ $n^2(q-1)^2(n-1)(q-n+2)$
$2n-2$ $n^2(q-1)^2(q-n+2)^2+n(n-1)(q-1)^2[(q-1)^2-(n-3)(q-1)+C_{n-2}^2]$
$2n-1$ $2n(q-n+2)(q-1)^4-2n(n-3)(q-n+2)(q-1)^3$
$+n(n-2)(n-3)(q-n+2)(q-1)^2$
$2n$ $(q-1)^6+(n-3)^2(q-1)^4+\frac {(n-2)^2(n-3)^2}4 (q-1)^2+(n-2)(n-3)(q-1)^4$
$-2(n-3)(q-1)^5-(n-2)(n-3)^2(q-1)^3$
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