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February  2021, 15(1): 1-8. doi: 10.3934/amc.2020038

New bounds on the minimum distance of cyclic codes

San Ling and  Buket Özkaya ,

School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

* Corresponding author

Received  December 2018 Revised  June 2019 Published  February 2021 Early access  November 2019

Fund Project: The authors are supported by NTU Research Grant M4080456

Two bounds on the minimum distance of cyclic codes are proposed. The first one generalizes the Roos bound by embedding the given cyclic code into a cyclic product code. The second bound also uses a second cyclic code, namely the non-zero-locator code, but is not directly related to cyclic product codes and it generalizes a special case of the Roos bound.

Keywords: Cyclic codes, product code, minimum distance bound, Roos bound.
Mathematics Subject Classification: Primary: 94B15, 94B65; Secondary: 11T71.
Citation: San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038
References:
[1]

R. C. Bose and D. K. R. Chaudhuri, On a class of error correcting binary group code, Information and Control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[2]

H. O. Burton and E. J. Weldon, Cyclic product codes, IEEE Trans. Information Theory, 11 (1965), 433-439.  doi: 10.1109/tit.1965.1053802.  Google Scholar

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C. Hartmann and K. Tzeng, Generalizations of the BCH bound, Information and Control, 20 (1972), 489-498.  doi: 10.1016/S0019-9958(72)90887-X.  Google Scholar

[4]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[5]

S. Lin and E. J. Weldon, Further results on cyclic product codes, IEEE Trans. Information Theory, 16 (1970), 452-459.  doi: 10.1109/tit.1970.1054491.  Google Scholar

[6]

C. Roos, A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound, J. Combin. Theory Ser. A, 33 (1982), 229-232.  doi: 10.1016/0097-3165(82)90014-0.  Google Scholar

[7]

C. Roos, A new lower bound for the minimum distance of a cyclic code, IEEE Trans. Inform. Theory, 29 (1983), 330-332.  doi: 10.1109/TIT.1983.1056672.  Google Scholar

[8]

A. Zeh and S. V. Bezzateev, A new bound on the minimum distance of cyclic codes using small-minimum-distance cyclic codes, Des. Codes Cryptogr., 71 (2014), 229-246.  doi: 10.1007/s10623-012-9721-3.  Google Scholar

[9]

A. Zeh, A. Wachter-Zeh, M. Gadouleau and S. V. Bezzateev, Generalizing bounds on the minimum distance of cyclic codes using cyclic product codes, Proc. IEEE ISIT, Istanbul, Turkey, 2013,126–130. doi: 10.1109/ISIT.2013.6620201.  Google Scholar

show all references

References:
[1]

R. C. Bose and D. K. R. Chaudhuri, On a class of error correcting binary group code, Information and Control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[2]

H. O. Burton and E. J. Weldon, Cyclic product codes, IEEE Trans. Information Theory, 11 (1965), 433-439.  doi: 10.1109/tit.1965.1053802.  Google Scholar

[3]

C. Hartmann and K. Tzeng, Generalizations of the BCH bound, Information and Control, 20 (1972), 489-498.  doi: 10.1016/S0019-9958(72)90887-X.  Google Scholar

[4]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[5]

S. Lin and E. J. Weldon, Further results on cyclic product codes, IEEE Trans. Information Theory, 16 (1970), 452-459.  doi: 10.1109/tit.1970.1054491.  Google Scholar

[6]

C. Roos, A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound, J. Combin. Theory Ser. A, 33 (1982), 229-232.  doi: 10.1016/0097-3165(82)90014-0.  Google Scholar

[7]

C. Roos, A new lower bound for the minimum distance of a cyclic code, IEEE Trans. Inform. Theory, 29 (1983), 330-332.  doi: 10.1109/TIT.1983.1056672.  Google Scholar

[8]

A. Zeh and S. V. Bezzateev, A new bound on the minimum distance of cyclic codes using small-minimum-distance cyclic codes, Des. Codes Cryptogr., 71 (2014), 229-246.  doi: 10.1007/s10623-012-9721-3.  Google Scholar

[9]

A. Zeh, A. Wachter-Zeh, M. Gadouleau and S. V. Bezzateev, Generalizing bounds on the minimum distance of cyclic codes using cyclic product codes, Proc. IEEE ISIT, Istanbul, Turkey, 2013,126–130. doi: 10.1109/ISIT.2013.6620201.  Google Scholar

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