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Further improvement of factoring $ N = p^r q^s$ with partial known bits
The weight distribution of a class of p-ary cyclic codes and their applications
School of Mathematics, Hefei University of Technology, Hefei 230601, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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, 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. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Workshop Inf. Theory, Molle, Sweden, (1993), 276-279. |
[23] |
J. L. Massey,
Some applications of coding theory, Cryptography, codes and Ciphers: Cryptography and Coding IV, (1995), 33-47.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
Z. Zhou and C. Ding,
Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.
|
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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, 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. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. L. Massey, Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian Workshop Inf. Theory, Molle, Sweden, (1993), 276-279. |
[23] |
J. L. Massey,
Some applications of coding theory, Cryptography, codes and Ciphers: Cryptography and Coding IV, (1995), 33-47.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
Z. Zhou and C. Ding,
Seven classes of three-weight cyclic codes, IEEE Trans. Commun., 61 (2013), 4120-4126.
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