May  2016, 10(2): 437-457. doi: 10.3934/amc.2016017

A class of $p$-ary cyclic codes and their weight enumerators

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China

Received  September 2014 Revised  October 2015 Published  April 2016

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime, and $m$ is a positive integer. Let $h_1(x)$ and $h_2(x)$ be minimal polynomials of $-\pi^{-1}$ and $\pi^{-\frac{p^k+1}{2}}$ over $\mathbb{F}_p$, respectively, where $\pi $ is a primitive element of $\mathbb{F}_{p^m}$, and $k$ is a positive integer such that $\frac{m}{\gcd(m,k)}\geq 3$. In [23], Zhou et al. obtained the weight distribution of a class of cyclic codes over $\mathbb{F}_p$ with parity-check polynomial $h_1(x)h_2(x)$ in the following two cases:
    • $k$ is even and $\gcd(m,k)$ is odd;
    • $\frac{m}{\gcd(m,k)}$ and $\frac{k}{\gcd(m,k)}$ are both odd. In this paper, we further investigate this class of cyclic codes over $\mathbb{F}_p$ in other cases. We determine the weight distribution of this class of cyclic codes.
Citation: Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
References:
[1]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes,, IEEE Trans. Inf. Theory, 21 (1975), 575.   Google Scholar

[2]

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.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[3]

C. Ding and J. Yang, Hamming weight in irrecducible codes,, Discrete Math., 313 (2013), 434.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[4]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes,, Finite Fields Appl., 14 (2008), 390.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[5]

T. Feng, On cyclic codes of length $2^{2^r}-1$ with two zeros whose dual codes have three weights,, Des. Codes Crypt., 62 (2012), 253.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[6]

R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1983).   Google Scholar

[7]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function,, IEEE Trans. Inf. Theory, 54 (2008), 5345.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[8]

J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes,, IEEE Trans. Inf. Theory, 54 (2008), 5332.  doi: 10.1109/TIT.2008.2006424.  Google Scholar

[9]

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.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[10]

F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1997).   Google Scholar

[11]

A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes,, IEEE Trans. Inf. Theory, 56 (2010), 2568.  doi: 10.1109/TIT.2010.2046201.  Google Scholar

[12]

G. Vega, The weight distribution of an extended class of reducible cyclic codes,, IEEE Trans. Inf. Theory, 58 (2012), 4862.  doi: 10.1109/TIT.2012.2193376.  Google Scholar

[13]

G. Vega and J. Wolfmann, New classes of $2$-weight cyclic codes,, Des. Codes Crypt., 42 (2007), 327.  doi: 10.1007/s10623-007-9038-9.  Google Scholar

[14]

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.  doi: 10.1109/TIT.2012.2210386.  Google Scholar

[15]

M. Xiong, The weight distributions of a class of cyclic codes,, Finite Fields Appl., 18 (2012), 933.  doi: 10.1016/j.ffa.2012.06.001.  Google Scholar

[16]

M. Xiong, The weight distributions of a class of cyclic codes II,, Des. Codes Crypt., 72 (2014), 511.  doi: 10.1007/s10623-012-9785-0.  Google Scholar

[17]

M. Xiong, The weight distributions of a class of cyclic codes III,, Finite Fields Appl., 21 (2013), 84.  doi: 10.1016/j.ffa.2013.01.004.  Google Scholar

[18]

L. Yu and H. Liu, The weight distribution of a family of p-ary cyclic codes,, Des. Codes Crypt., 78 (2016), 731.  doi: 10.1007/s10623-014-0029-3.  Google Scholar

[19]

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.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[20]

X. Zeng, J. Shan and L. Hu, A triple-error-correcting cyclic code from the Gold and Kasami-Welch APN power functions,, Finite Fields Appl., 18 (2012), 70.  doi: 10.1016/j.ffa.2011.06.005.  Google Scholar

[21]

D. Zheng, X. Wang, L. Yu and H. Liu, The weight enumerators of several classes of $p$-ary cyclic codes,, Discrete Math., 338 (2015), 1264.  doi: 10.1016/j.disc.2015.02.005.  Google Scholar

[22]

D. Zheng, X. Wang, X. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes,, Des. Codes Crypt., 75 (2015), 263.  doi: 10.1007/s10623-013-9908-2.  Google Scholar

[23]

Z. Zhou and C. Ding, A class of three-weight cyclic codes,, Finite Fields Appl., 25 (2014), 79.  doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

[24]

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.  doi: 10.1109/TIT.2013.2267722.  Google Scholar

show all references

References:
[1]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes,, IEEE Trans. Inf. Theory, 21 (1975), 575.   Google Scholar

[2]

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.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[3]

C. Ding and J. Yang, Hamming weight in irrecducible codes,, Discrete Math., 313 (2013), 434.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[4]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes,, Finite Fields Appl., 14 (2008), 390.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[5]

T. Feng, On cyclic codes of length $2^{2^r}-1$ with two zeros whose dual codes have three weights,, Des. Codes Crypt., 62 (2012), 253.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[6]

R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1983).   Google Scholar

[7]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function,, IEEE Trans. Inf. Theory, 54 (2008), 5345.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[8]

J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes,, IEEE Trans. Inf. Theory, 54 (2008), 5332.  doi: 10.1109/TIT.2008.2006424.  Google Scholar

[9]

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.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[10]

F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1997).   Google Scholar

[11]

A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes,, IEEE Trans. Inf. Theory, 56 (2010), 2568.  doi: 10.1109/TIT.2010.2046201.  Google Scholar

[12]

G. Vega, The weight distribution of an extended class of reducible cyclic codes,, IEEE Trans. Inf. Theory, 58 (2012), 4862.  doi: 10.1109/TIT.2012.2193376.  Google Scholar

[13]

G. Vega and J. Wolfmann, New classes of $2$-weight cyclic codes,, Des. Codes Crypt., 42 (2007), 327.  doi: 10.1007/s10623-007-9038-9.  Google Scholar

[14]

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.  doi: 10.1109/TIT.2012.2210386.  Google Scholar

[15]

M. Xiong, The weight distributions of a class of cyclic codes,, Finite Fields Appl., 18 (2012), 933.  doi: 10.1016/j.ffa.2012.06.001.  Google Scholar

[16]

M. Xiong, The weight distributions of a class of cyclic codes II,, Des. Codes Crypt., 72 (2014), 511.  doi: 10.1007/s10623-012-9785-0.  Google Scholar

[17]

M. Xiong, The weight distributions of a class of cyclic codes III,, Finite Fields Appl., 21 (2013), 84.  doi: 10.1016/j.ffa.2013.01.004.  Google Scholar

[18]

L. Yu and H. Liu, The weight distribution of a family of p-ary cyclic codes,, Des. Codes Crypt., 78 (2016), 731.  doi: 10.1007/s10623-014-0029-3.  Google Scholar

[19]

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.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[20]

X. Zeng, J. Shan and L. Hu, A triple-error-correcting cyclic code from the Gold and Kasami-Welch APN power functions,, Finite Fields Appl., 18 (2012), 70.  doi: 10.1016/j.ffa.2011.06.005.  Google Scholar

[21]

D. Zheng, X. Wang, L. Yu and H. Liu, The weight enumerators of several classes of $p$-ary cyclic codes,, Discrete Math., 338 (2015), 1264.  doi: 10.1016/j.disc.2015.02.005.  Google Scholar

[22]

D. Zheng, X. Wang, X. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes,, Des. Codes Crypt., 75 (2015), 263.  doi: 10.1007/s10623-013-9908-2.  Google Scholar

[23]

Z. Zhou and C. Ding, A class of three-weight cyclic codes,, Finite Fields Appl., 25 (2014), 79.  doi: 10.1016/j.ffa.2013.08.005.  Google Scholar

[24]

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.  doi: 10.1109/TIT.2013.2267722.  Google Scholar

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