# American Institute of Mathematical Sciences

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

## 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

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 $n>4$.

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