American Institute of Mathematical Sciences

Advanced
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$ $p-1$
Table Options

Download as excel

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
[1]

Ryutaroh Matsumoto. Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields. Advances in Mathematics of Communications, 2019, 13 (1) : 1-10. doi: 10.3934/amc.2019001
[2]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[3]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[4]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[5]

Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005
[6]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
[7]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[8]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[9]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191
[10]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[11]

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
[12]

I-Lin Wang, Ju-Chun Lin. A compaction scheme and generator for distribution networks. Journal of Industrial & Management Optimization, 2016, 12 (1) : 117-140. doi: 10.3934/jimo.2016.12.117
[13]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023
[14]

Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011
[15]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[16]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[17]

Thomas Gauthier, Gabriel Vigny. Distribution of postcritically finite polynomials Ⅱ: Speed of convergence. Journal of Modern Dynamics, 2017, 11: 57-98. doi: 10.3934/jmd.2017004
[18]

Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277
[19]

Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341
[20]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (94)
  • HTML views (323)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences