# American Institute of Mathematical Sciences

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), 575-576. [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), 8000-8006. doi: 10.1109/TIT.2011.2165314. [3] C. Ding and J. Yang, Hamming weight in irrecducible codes, Discrete Math., 313 (2013), 434-446. doi: 10.1016/j.disc.2012.11.009. [4] K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003. [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), 253-258. doi: 10.1007/s10623-011-9514-0. [6] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1983. [7] J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353. doi: 10.1109/TIT.2008.2006394. [8] J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424. [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), 397-402. doi: 10.1109/TIT.2010.2090272. [10] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1997. [11] A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 2568-2570. doi: 10.1109/TIT.2010.2046201. [12] 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. [13] G. Vega and J. Wolfmann, New classes of $2$-weight cyclic codes, Des. Codes Crypt., 42 (2007), 327-334. doi: 10.1007/s10623-007-9038-9. [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), 7253-7259. doi: 10.1109/TIT.2012.2210386. [15] 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. [16] M. Xiong, The weight distributions of a class of cyclic codes II, Des. Codes Crypt., 72 (2014), 511-528. doi: 10.1007/s10623-012-9785-0. [17] M. Xiong, The weight distributions of a class of cyclic codes III, Finite Fields Appl., 21 (2013), 84-96. doi: 10.1016/j.ffa.2013.01.004. [18] L. Yu and H. Liu, The weight distribution of a family of p-ary cyclic codes, Des. Codes Crypt., 78 (2016), 731-745. doi: 10.1007/s10623-014-0029-3. [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), 56-73. doi: 10.1016/j.ffa.2009.12.001. [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), 70-92. doi: 10.1016/j.ffa.2011.06.005. [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), 1264-1276. doi: 10.1016/j.disc.2015.02.005. [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), 263-275. doi: 10.1007/s10623-013-9908-2. [23] 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. [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), 6674-6682. doi: 10.1109/TIT.2013.2267722.

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##### References:
 [1] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576. [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), 8000-8006. doi: 10.1109/TIT.2011.2165314. [3] C. Ding and J. Yang, Hamming weight in irrecducible codes, Discrete Math., 313 (2013), 434-446. doi: 10.1016/j.disc.2012.11.009. [4] K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409. doi: 10.1016/j.ffa.2007.03.003. [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), 253-258. doi: 10.1007/s10623-011-9514-0. [6] R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1983. [7] J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353. doi: 10.1109/TIT.2008.2006394. [8] J. Luo and K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344. doi: 10.1109/TIT.2008.2006424. [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), 397-402. doi: 10.1109/TIT.2010.2090272. [10] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1997. [11] A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 2568-2570. doi: 10.1109/TIT.2010.2046201. [12] 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. [13] G. Vega and J. Wolfmann, New classes of $2$-weight cyclic codes, Des. Codes Crypt., 42 (2007), 327-334. doi: 10.1007/s10623-007-9038-9. [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), 7253-7259. doi: 10.1109/TIT.2012.2210386. [15] 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. [16] M. Xiong, The weight distributions of a class of cyclic codes II, Des. Codes Crypt., 72 (2014), 511-528. doi: 10.1007/s10623-012-9785-0. [17] M. Xiong, The weight distributions of a class of cyclic codes III, Finite Fields Appl., 21 (2013), 84-96. doi: 10.1016/j.ffa.2013.01.004. [18] L. Yu and H. Liu, The weight distribution of a family of p-ary cyclic codes, Des. Codes Crypt., 78 (2016), 731-745. doi: 10.1007/s10623-014-0029-3. [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), 56-73. doi: 10.1016/j.ffa.2009.12.001. [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), 70-92. doi: 10.1016/j.ffa.2011.06.005. [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), 1264-1276. doi: 10.1016/j.disc.2015.02.005. [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), 263-275. doi: 10.1007/s10623-013-9908-2. [23] 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. [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), 6674-6682. doi: 10.1109/TIT.2013.2267722.
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