-
Previous Article
A construction of $ p $-ary linear codes with two or three weights
- AMC Home
- This Issue
- Next Article
New bounds on the minimum distance of cyclic codes
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 |
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.
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. |
| [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. |
| [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. |
| [4] |
A. Hocquenghem,
Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
A. Hocquenghem,
Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [1] |
José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018 |
| [2] |
Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175 |
| [3] |
Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021043 |
| [4] |
Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055 |
| [5] |
Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165 |
| [6] |
Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275 |
| [7] |
Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65 |
| [8] |
J. De Beule, K. Metsch, L. Storme. Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound. Advances in Mathematics of Communications, 2008, 2 (3) : 261-272. doi: 10.3934/amc.2008.2.261 |
| [9] |
Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1) : 35-52. doi: 10.3934/amc.2014.8.35 |
| [10] |
Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83 |
| [11] |
Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259 |
| [12] |
Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021210 |
| [13] |
Gang Wang, Deng-Ming Xu, Fang-Wei Fu. Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021065 |
| [14] |
Mikko Kaasalainen. Dynamical tomography of gravitationally bound systems. Inverse Problems and Imaging, 2008, 2 (4) : 527-546. doi: 10.3934/ipi.2008.2.527 |
| [15] |
Z.G. Feng, K.L. Teo, Y. Zhao. Branch and bound method for sensor scheduling in discrete time. Journal of Industrial and Management Optimization, 2005, 1 (4) : 499-512. doi: 10.3934/jimo.2005.1.499 |
| [16] |
Marcin Dumnicki, Łucja Farnik, Halszka Tutaj-Gasińska. Asymptotic Hilbert polynomial and a bound for Waldschmidt constants. Electronic Research Announcements, 2016, 23: 8-18. doi: 10.3934/era.2016.23.002 |
| [17] |
Miklós Horváth, Márton Kiss. A bound for ratios of eigenvalues of Schrodinger operators on the real line. Conference Publications, 2005, 2005 (Special) : 403-409. doi: 10.3934/proc.2005.2005.403 |
| [18] |
John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7. |
| [19] |
Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294 |
| [20] |
Yuntao Zang. An upper bound of the measure-theoretical entropy. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022052 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]