November  2016, 10(4): 707-723. doi: 10.3934/amc.2016036

Cyclic codes from two-prime generalized cyclotomic sequences of order 6

1. 

College of Sciences, China University of Petroleum, 66 Changjiang Xilu, Qingdao, Shandong 266580

2. 

College of Science, China University of Petroleum, Qingdao, Shandong 266580, China

Received  July 2014 Published  November 2016

Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\mathrm{GF}(q)$ and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds on the minimum distance for some other cyclic codes.
Citation: Tongjiang Yan, Yanyan Liu, Yuhua Sun. Cyclic codes from two-prime generalized cyclotomic sequences of order 6. Advances in Mathematics of Communications, 2016, 10 (4) : 707-723. doi: 10.3934/amc.2016036
References:
[1]

E. Betti and M. Sala, A new bound for the minimum distance of a cyclic code from its defining set,, IEEE Trans. Inform. Theory, 52 (2006), 3700.  doi: 10.1109/TIT.2006.876240.  Google Scholar

[2]

Z. Chen and X. Du, Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length $pq$,, Front. Computer Sci. China, 4 (2010), 529.   Google Scholar

[3]

T. Cusick, C. Ding and A. Renvall, Stream ciphers and number theory,, North-Holland Math. Library, 55 (2004), 198.   Google Scholar

[4]

C. Ding, Cyclic codes from APN and planar functions,, preprint, ().   Google Scholar

[5]

C. Ding, Cyclic codes from cyclotomic sequences of order four,, Finite Fields Appl., 23 (2012), 8.  doi: 10.1016/j.ffa.2013.03.006.  Google Scholar

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C. Ding, Cyclic codes from dickson polynomials,, preprint, ().   Google Scholar

[7]

C. Ding, Cyclic codes from the two-prime sequences,, IEEE Trans. Inform. Theory, 58 (2012), 3881.  doi: 10.1109/TIT.2012.2189549.  Google Scholar

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C. Ding, Cyclotomic constructions of cyclic codes with length being the product of two primes,, IEEE Trans. Inform. Theory, 58 (2012), 2231.  doi: 10.1109/TIT.2011.2176915.  Google Scholar

[9]

C. Ding and S. Ling, A $q$-polynomial approach to cyclic codes,, Finite Fields Appl., 20 (2013), 1.  doi: 10.1016/j.ffa.2012.12.005.  Google Scholar

[10]

C. Ding, Y. Liu and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros,, IEEE Trans. Inform. Theory, 57 (2011), 8000.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[11]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes,, IEEE Trans. Inform. Theory, 313 (2013), 434.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[12]

M. van Eupen and J. H. van Lint, On the minimum distance of ternary cyclic codes,, IEEE Trans. Inform. Theory, 39 (1993), 409.  doi: 10.1109/18.212272.  Google Scholar

[13]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[14]

J. H. van Lint and R. M. Wilson, On the minimum distance of cyclic codes,, IEEE Trans. Inform. Theory, 32 (1986), 23.  doi: 10.1109/TIT.1986.1057134.  Google Scholar

[15]

C. Ma, L. Zeng and Y. Liu et al., The weight enumerator of a class of cyclic codes,, IEEE Trans. Inform. Theory, 57 (2011), 397.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[16]

Y. Sun, T. Yan and H. Li, Cyclic code from the first class Whiteman's generalized cyclotomic sequence with order 4,, preprint, ().   Google Scholar

[17]

A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107.   Google Scholar

[18]

T. Yan, X. Du, G. Xiao and X. Huang, Linear complexity of binary Whiteman generalized cyclotomic sequences of order $2k$,, Inform. Sci., 179 (2009), 1019.  doi: 10.1016/j.ins.2008.11.006.  Google Scholar

[19]

T. Yan, B. Huang and G. Xiao, Cryptographic properties of some binary generalized cyclotomic sequences with the length $p^2$,, Inform. Sci., 178 (2008), 1078.  doi: 10.1016/j.ins.2007.02.040.  Google Scholar

[20]

J. Yang, M. Xiong and C. Ding, Weight distribution of a class of cyclic codes with arbitrary number of zeros,, IEEE Trans. Inform. Theory, 9 (2013), 5985.  doi: 10.1109/TIT.2013.2266731.  Google Scholar

[21]

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

[22]

Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators,, IEEE Trans. Inform. Theory, 10 (2013), 6674.  doi: 10.1109/TIT.2013.2267722.  Google Scholar

[23]

Z. Zhou, A. Zhang and C. Ding, The weight enumerator of three families of cyclic codes,, IEEE Trans. Inform. Theory, 9 (2013), 6002.  doi: 10.1109/TIT.2013.2262095.  Google Scholar

show all references

References:
[1]

E. Betti and M. Sala, A new bound for the minimum distance of a cyclic code from its defining set,, IEEE Trans. Inform. Theory, 52 (2006), 3700.  doi: 10.1109/TIT.2006.876240.  Google Scholar

[2]

Z. Chen and X. Du, Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length $pq$,, Front. Computer Sci. China, 4 (2010), 529.   Google Scholar

[3]

T. Cusick, C. Ding and A. Renvall, Stream ciphers and number theory,, North-Holland Math. Library, 55 (2004), 198.   Google Scholar

[4]

C. Ding, Cyclic codes from APN and planar functions,, preprint, ().   Google Scholar

[5]

C. Ding, Cyclic codes from cyclotomic sequences of order four,, Finite Fields Appl., 23 (2012), 8.  doi: 10.1016/j.ffa.2013.03.006.  Google Scholar

[6]

C. Ding, Cyclic codes from dickson polynomials,, preprint, ().   Google Scholar

[7]

C. Ding, Cyclic codes from the two-prime sequences,, IEEE Trans. Inform. Theory, 58 (2012), 3881.  doi: 10.1109/TIT.2012.2189549.  Google Scholar

[8]

C. Ding, Cyclotomic constructions of cyclic codes with length being the product of two primes,, IEEE Trans. Inform. Theory, 58 (2012), 2231.  doi: 10.1109/TIT.2011.2176915.  Google Scholar

[9]

C. Ding and S. Ling, A $q$-polynomial approach to cyclic codes,, Finite Fields Appl., 20 (2013), 1.  doi: 10.1016/j.ffa.2012.12.005.  Google Scholar

[10]

C. Ding, Y. Liu and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros,, IEEE Trans. Inform. Theory, 57 (2011), 8000.  doi: 10.1109/TIT.2011.2165314.  Google Scholar

[11]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes,, IEEE Trans. Inform. Theory, 313 (2013), 434.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[12]

M. van Eupen and J. H. van Lint, On the minimum distance of ternary cyclic codes,, IEEE Trans. Inform. Theory, 39 (1993), 409.  doi: 10.1109/18.212272.  Google Scholar

[13]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[14]

J. H. van Lint and R. M. Wilson, On the minimum distance of cyclic codes,, IEEE Trans. Inform. Theory, 32 (1986), 23.  doi: 10.1109/TIT.1986.1057134.  Google Scholar

[15]

C. Ma, L. Zeng and Y. Liu et al., The weight enumerator of a class of cyclic codes,, IEEE Trans. Inform. Theory, 57 (2011), 397.  doi: 10.1109/TIT.2010.2090272.  Google Scholar

[16]

Y. Sun, T. Yan and H. Li, Cyclic code from the first class Whiteman's generalized cyclotomic sequence with order 4,, preprint, ().   Google Scholar

[17]

A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107.   Google Scholar

[18]

T. Yan, X. Du, G. Xiao and X. Huang, Linear complexity of binary Whiteman generalized cyclotomic sequences of order $2k$,, Inform. Sci., 179 (2009), 1019.  doi: 10.1016/j.ins.2008.11.006.  Google Scholar

[19]

T. Yan, B. Huang and G. Xiao, Cryptographic properties of some binary generalized cyclotomic sequences with the length $p^2$,, Inform. Sci., 178 (2008), 1078.  doi: 10.1016/j.ins.2007.02.040.  Google Scholar

[20]

J. Yang, M. Xiong and C. Ding, Weight distribution of a class of cyclic codes with arbitrary number of zeros,, IEEE Trans. Inform. Theory, 9 (2013), 5985.  doi: 10.1109/TIT.2013.2266731.  Google Scholar

[21]

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

[22]

Z. Zhou, C. Ding, J. Luo and A. Zhang, A family of five-weight cyclic codes and their weight enumerators,, IEEE Trans. Inform. Theory, 10 (2013), 6674.  doi: 10.1109/TIT.2013.2267722.  Google Scholar

[23]

Z. Zhou, A. Zhang and C. Ding, The weight enumerator of three families of cyclic codes,, IEEE Trans. Inform. Theory, 9 (2013), 6002.  doi: 10.1109/TIT.2013.2262095.  Google Scholar

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