doi: 10.3934/amc.2020038

New bounds on the minimum distance of cyclic codes

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

* Corresponding author

Received  December 2018 Revised  June 2019 Published  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.

Citation: San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 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

[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

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

[1]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[2]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104

[3]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118

[4]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

[5]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (131)
  • HTML views (460)
  • Cited by (0)

Other articles
by authors

[Back to Top]