American Institute of Mathematical Sciences

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February  2019, 13(1): 137-156. doi: 10.3934/amc.2019008

The weight distribution of a class of p-ary cyclic codes and their applications

Lanqiang Li , Shixin Zhu , and  Li Liu

School of Mathematics, Hefei University of Technology, Hefei 230601, China

* Corresponding author: Shixin Zhu

Received  May 2018 Revised  August 2018 Published  December 2018

Fund Project: This research is supported in part by the National Natural Science Foundation of China under Project 61572168, Project 61772168 and Project 11501156, the Natural Science Foundation of Anhui Province under Grant 1808085MA15 and the Key University Science Research Project of Anhui Province under Grant KJ2018A0497

Cyclic codes over finite field have been studied for decades due to their wide applications in communication and storage systems. However their weight distributions are known only in a few cases. In this paper, we investigate a class of $ p$-ary cyclic codes whose duals have three zeros, where $ p$ is an odd prime. The weight distributions of the class of cyclic codes for all distinct cases are determined explicitly. The results indicate that these codes contain five-weight codes, seven-weight codes and eleven-weight codes. Some of these codes are optimal. Moreover, the covering structures of the class of codes are considered and being used to construct secret sharing schemes.

Keywords: Finite field, cyclic code, weight distribution, covering structure, secret sharing scheme.
Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.
Citation: Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008
References:
[1]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[3]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[4]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[5]

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.  Google Scholar

[6]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.  doi: 10.1109/TIT.2013.2281205.  Google Scholar

[7]

C. Ding, D. Kohel and S. Ling, Secret sharing with a class of ternary codes, Theo. Comput. Sci., 246 (2000), 285-298.  doi: 10.1016/S0304-3975(00)00207-3.  Google Scholar

[8]

K. Feng and J. Luo, Value distributions of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041.  doi: 10.1109/TIT.2007.903153.  Google Scholar

[9]

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.  Google Scholar

[10]

T. Feng, On cyclic codes of length $ 2^{2^r}-1$ with two zeros whose dual code have three weights, Des. Codes Cryptogr., 62 (2012), 253-258.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[11]

C. Li, Q. Yue and F.-W. Fu, Complete weight enumerators of some cyclic codes, Des. Codes Cryptogr., 80 (2016), 295-315.  doi: 10.1007/s10623-015-0091-5.  Google Scholar

[12]

C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.  doi: 10.1109/TIT.2014.2329694.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Li, S. Ling and L. Qu, On the covering structures of two classes of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 55 (2009), 70-82.  doi: 10.1109/TIT.2008.2008145.  Google Scholar

[16]

R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.  Google Scholar

[17]

Y. Liu and H. Yan, A class of five-weight cyclic codes and their weight distribution, Des. Codes Cryptogr., 79 (2016), 353-366.  doi: 10.1007/s10623-015-0056-8.  Google Scholar

[18]

X. Liu and Y. Luo, The weight distributions of some cyclic codes with three or four nonzeros over $ F_3$, Des. Codes Cryptogr., 73 (2014), 747-768.  doi: 10.1007/s10623-013-9824-5.  Google Scholar

[19]

J. Luo and K. Feng, Cyclic codes and sequences from generalized CoulterMatthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[20]

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.  Google Scholar

[21]

F. E. B. Martinez and C. R. G. Vergara, Weight enumerator of some irreducible cyclic codes, Des. Codes Cryptogr., 78 (2016), 703-712.  doi: 10.1007/s10623-014-0026-6.  Google Scholar

[22]

J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Workshop Inf. Theory, Molle, Sweden, (1993), 276-279. Google Scholar

[23]

J. L. Massey, Some applications of coding theory, Cryptography, codes and Ciphers: Cryptography and Coding IV, (1995), 33-47.   Google Scholar

[24]

K. U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory, J. Algebraic Comb., 42 (2015), 635-670.  doi: 10.1007/s10801-015-0595-0.  Google Scholar

[25]

Z. Shi and F.-W. Fu, A complete weight enumerators of some irreducible cyclic codes, Discrete Applied Math., 219 (2017), 182-192.  doi: 10.1016/j.dam.2016.11.008.  Google Scholar

[26]

M. Xiong, N. Li, Z. Zhou and C. Ding, Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes, Des. Codes Cryptogr., 78 (2016), 713-730.  doi: 10.1007/s10623-014-0027-5.  Google Scholar

[27]

M. Xiong, The weight distributions of a class of cyclic codes Ⅱ, Des. Codes Cryptogr., 72 (2014), 511-528.  doi: 10.1007/s10623-012-9785-0.  Google Scholar

[28]

H. Yan and C. Liu, Two classes of cyclic codes and their weight enumerator, Des. Codes Cryptogr., 81 (2016), 1-9.  doi: 10.1007/s10623-015-0125-z.  Google Scholar

[29]

S. Yang, Z. Yao and C. Zhao, The weight distributions of two classes of pary cyclic codes with few weights, Finite Fields Appl., 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.  Google Scholar

[30]

J. Yang, M. Xiong, C. Ding and J. Luo, 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.  Google Scholar

[31]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar

[32]

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.  Google Scholar

[33]

X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of pary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[34]

D. Zheng, X. Wang, H. Hu and X. Zeng, The weight distributions of two classes of pary cyclic codes, Finite Fields Appl., 29 (2014), 202-224.  doi: 10.1016/j.ffa.2014.05.001.  Google Scholar

[35]

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.  Google Scholar

[36]

Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.   Google Scholar

show all references

References:
[1]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.  Google Scholar

[2]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.  Google Scholar

[3]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[4]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[5]

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.  Google Scholar

[6]

C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59 (2013), 7940-7946.  doi: 10.1109/TIT.2013.2281205.  Google Scholar

[7]

C. Ding, D. Kohel and S. Ling, Secret sharing with a class of ternary codes, Theo. Comput. Sci., 246 (2000), 285-298.  doi: 10.1016/S0304-3975(00)00207-3.  Google Scholar

[8]

K. Feng and J. Luo, Value distributions of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inf. Theory, 53 (2007), 3035-3041.  doi: 10.1109/TIT.2007.903153.  Google Scholar

[9]

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.  Google Scholar

[10]

T. Feng, On cyclic codes of length $ 2^{2^r}-1$ with two zeros whose dual code have three weights, Des. Codes Cryptogr., 62 (2012), 253-258.  doi: 10.1007/s10623-011-9514-0.  Google Scholar

[11]

C. Li, Q. Yue and F.-W. Fu, Complete weight enumerators of some cyclic codes, Des. Codes Cryptogr., 80 (2016), 295-315.  doi: 10.1007/s10623-015-0091-5.  Google Scholar

[12]

C. Li, N. Li, T. Helleseth and C. Ding, The weight distributions of several classes of cyclic codes from APN monomials, IEEE Trans. Inf. Theory, 60 (2014), 4710-4721.  doi: 10.1109/TIT.2014.2329694.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Li, S. Ling and L. Qu, On the covering structures of two classes of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 55 (2009), 70-82.  doi: 10.1109/TIT.2008.2008145.  Google Scholar

[16]

R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.  Google Scholar

[17]

Y. Liu and H. Yan, A class of five-weight cyclic codes and their weight distribution, Des. Codes Cryptogr., 79 (2016), 353-366.  doi: 10.1007/s10623-015-0056-8.  Google Scholar

[18]

X. Liu and Y. Luo, The weight distributions of some cyclic codes with three or four nonzeros over $ F_3$, Des. Codes Cryptogr., 73 (2014), 747-768.  doi: 10.1007/s10623-013-9824-5.  Google Scholar

[19]

J. Luo and K. Feng, Cyclic codes and sequences from generalized CoulterMatthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[20]

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.  Google Scholar

[21]

F. E. B. Martinez and C. R. G. Vergara, Weight enumerator of some irreducible cyclic codes, Des. Codes Cryptogr., 78 (2016), 703-712.  doi: 10.1007/s10623-014-0026-6.  Google Scholar

[22]

J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Workshop Inf. Theory, Molle, Sweden, (1993), 276-279. Google Scholar

[23]

J. L. Massey, Some applications of coding theory, Cryptography, codes and Ciphers: Cryptography and Coding IV, (1995), 33-47.   Google Scholar

[24]

K. U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory, J. Algebraic Comb., 42 (2015), 635-670.  doi: 10.1007/s10801-015-0595-0.  Google Scholar

[25]

Z. Shi and F.-W. Fu, A complete weight enumerators of some irreducible cyclic codes, Discrete Applied Math., 219 (2017), 182-192.  doi: 10.1016/j.dam.2016.11.008.  Google Scholar

[26]

M. Xiong, N. Li, Z. Zhou and C. Ding, Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes, Des. Codes Cryptogr., 78 (2016), 713-730.  doi: 10.1007/s10623-014-0027-5.  Google Scholar

[27]

M. Xiong, The weight distributions of a class of cyclic codes Ⅱ, Des. Codes Cryptogr., 72 (2014), 511-528.  doi: 10.1007/s10623-012-9785-0.  Google Scholar

[28]

H. Yan and C. Liu, Two classes of cyclic codes and their weight enumerator, Des. Codes Cryptogr., 81 (2016), 1-9.  doi: 10.1007/s10623-015-0125-z.  Google Scholar

[29]

S. Yang, Z. Yao and C. Zhao, The weight distributions of two classes of pary cyclic codes with few weights, Finite Fields Appl., 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.  Google Scholar

[30]

J. Yang, M. Xiong, C. Ding and J. Luo, 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.  Google Scholar

[31]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar

[32]

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.  Google Scholar

[33]

X. Zeng, L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of pary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[34]

D. Zheng, X. Wang, H. Hu and X. Zeng, The weight distributions of two classes of pary cyclic codes, Finite Fields Appl., 29 (2014), 202-224.  doi: 10.1016/j.ffa.2014.05.001.  Google Scholar

[35]

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.  Google Scholar

[36]

Z. Zhou and C. Ding, Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.   Google Scholar

Table Ⅰ.  Weight distribution of the cyclic code C for odd m in Theorem 3.1
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $ (p^m-1)(p^{m-1}+1)$
$(p-1)p^{m-1}-1$ $ (p^m-1)(p^{m-1}+1)(p-1)$
$(p-1)p^{m-1}-p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-1}{2}})(p-1) $
$ (p-1)p^{m-1}-p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}-p^{\frac{m-1}{2}}\big)(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-1}{2}})(p-1)$
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}+p^{\frac{m-1}{2}}\big)(p-1) $
$ p^{m}-1$ $p-1$
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $ (p^m-1)(p^{m-1}+1)$
$(p-1)p^{m-1}-1$ $ (p^m-1)(p^{m-1}+1)(p-1)$
$(p-1)p^{m-1}-p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-1}{2}})(p-1) $
$ (p-1)p^{m-1}-p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}-p^{\frac{m-1}{2}}\big)(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-1}{2}})(p-1)$
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}+p^{\frac{m-1}{2}}\big)(p-1) $
$ p^{m}-1$ $p-1$
Table Ⅱ.  Weight distribution of the cyclic code C for even m in Theorem 3.1
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $p^m-1 $
$(p-1)p^{m-1}-1$ $(p^m-1)(p-1) $
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}}) $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}})$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1)$
$ p^{m}-1$ $p-1 $
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $p^m-1 $
$(p-1)p^{m-1}-1$ $(p^m-1)(p-1) $
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}}) $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}})$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1)$
$ p^{m}-1$ $p-1 $
Table Ⅲ.  Weight distribution of the cyclic code C for even m in Theorem 3.2
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $(p^m-1)(1+p^{m-d}-p^{m-2d}) $
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}+ p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ p^{m}-1$ $p-1 $
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $(p^m-1)(1+p^{m-d}-p^{m-2d}) $
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}+ p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ p^{m}-1$ $p-1 $
Table Ⅳ.  Weight distribution of the cyclic code C for even m in Theorem 3.2
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $ (p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $(p^{m-1}+(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{(p^m-1)}{p^d+1}$
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ p^{m}-1$ $p-1 $
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $ (p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $(p^{m-1}+(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{(p^m-1)}{p^d+1}$
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ p^{m}-1$ $p-1 $
Table Ⅴ.  Weight distribution of the cyclic code C for even m in Corollary 1
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $(p^m-1) $
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p^d-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})(p^d-1)$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1) $
$ p^{m}-1$ $p-1 $
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p^{m-1}$ $(p^m-1) $
$(p-1)p^{m-1}-1$ $ (p-1)(p^m-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p^d-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})(p^d-1)$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1) $
$ p^{m}-1$ $p-1 $
Table Ⅵ.  Weight distribution of the cyclic code C for even m in Corollary 2
Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p$ $(p^2-1) $
$p(p-1)-1$ $ 2(p-1)(p^2-1)$
$(p-1)p-2$ $(p-1)^2(p^2-p-1) $
$ p^2-2$ $(p-1)(p^2-1) $
$ p^2-1$ $2(p-1) $
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Hamming Weight $i$ Frequency $A_i$
$0$ $1$
$(p-1)p$ $(p^2-1) $
$p(p-1)-1$ $ 2(p-1)(p^2-1)$
$(p-1)p-2$ $(p-1)^2(p^2-p-1) $
$ p^2-2$ $(p-1)(p^2-1) $
$ p^2-1$ $2(p-1) $
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