Cyclic and BCH codes whose minimum distance equals their maximum BCH bound

Previous Article
AMC Home
This Issue
Next Article
A class of $p$-ary cyclic codes and their weight enumerators

May 2016, 10(2): 459-474. doi: 10.3934/amc.2016018

Cyclic and BCH codes whose minimum distance equals their maximum BCH bound

José Joaquín Bernal ^1, , Diana H. Bueno-Carreño ^2, and Juan Jacobo Simón ^1,

1.	Departamento de Matemáticas, Universidad de Murcia, Spain, Spain
2.	Departamento de Ciencias Naturales y Matemáticas, Pontificia Universidad Javeriana seccional Cali, Colombia

Received October 2014 Revised September 2015 Published April 2016

Abstract
Related Papers

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form $x^n-1$. We apply our results to the study of those BCH codes $C$, with designed distance $\delta$, that have minimum distance $d(C)=\delta$. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.

Keywords: BCH codes, Cyclic codes, BCH bound, minimum distance., apparent distance.

Mathematics Subject Classification: 94B15, 94B6.

Citation: 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

References:

[1]	P. Camion, Abelian codes, MRC Tech. Sum. Rep. 1059, Univ. Wisconsin Madison, 1970.
[2]	P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 963-1063.
[3]	R. T. Chien and D. M. Choy, Algebraic generalization of BCH-Goppa-Helgert codes, IEEE Trans. Inf. Theory, 21 (1975), 70-79.
[4]	, GAP - Groups, Algorithms, Programming - a system for computational discrete algebra, http://www.gap-system.org/
[5]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.
[6]	F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.
[7]	J. H. Van Lint and R. M. Wilson, On the minimum distance of cyclic codes, IEEE Trans. Inf. Theory, 32 (1986), 23-40. doi: 10.1109/TIT.1986.1057134.

show all references

References:

[1]	P. Camion, Abelian codes, MRC Tech. Sum. Rep. 1059, Univ. Wisconsin Madison, 1970.
[2]	P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 963-1063.
[3]	R. T. Chien and D. M. Choy, Algebraic generalization of BCH-Goppa-Helgert codes, IEEE Trans. Inf. Theory, 21 (1975), 70-79.
[4]	, GAP - Groups, Algorithms, Programming - a system for computational discrete algebra, http://www.gap-system.org/
[5]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.
[6]	F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.
[7]	J. H. Van Lint and R. M. Wilson, On the minimum distance of cyclic codes, IEEE Trans. Inf. Theory, 32 (1986), 23-40. doi: 10.1109/TIT.1986.1057134.

[1]	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
[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]	Angelot Behajaina. On BCH split metacyclic codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021045
[4]	Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021043
[5]	Xinmei Huang, Qin Yue, Yansheng Wu, Xiaoping Shi. Ternary Primitive LCD BCH codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021014
[6]	Haode Yan, Hao Liu, Chengju Li, Shudi Yang. Parameters of LCD BCH codes with two lengths. Advances in Mathematics of Communications, 2018, 12 (3) : 579-594. doi: 10.3934/amc.2018034
[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]	Claude Carlet. Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022047
[9]	Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015
[10]	John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019
[11]	Carlos Munuera, Morgan Barbier. Wet paper codes and the dual distance in steganography. Advances in Mathematics of Communications, 2012, 6 (3) : 273-285. doi: 10.3934/amc.2012.6.273
[12]	Liqin Hu, Qin Yue, Fengmei Liu. Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3). Advances in Mathematics of Communications, 2014, 8 (3) : 297-312. doi: 10.3934/amc.2014.8.297
[13]	Andries E. Brouwer, Tuvi Etzion. Some new distance-4 constant weight codes. Advances in Mathematics of Communications, 2011, 5 (3) : 417-424. doi: 10.3934/amc.2011.5.417
[14]	Joaquim Borges, Josep Rifà, Victor A. Zinoviev. Families of nested completely regular codes and distance-regular graphs. Advances in Mathematics of Communications, 2015, 9 (2) : 233-246. doi: 10.3934/amc.2015.9.233
[15]	Diego Napp, Roxana Smarandache. Constructing strongly-MDS convolutional codes with maximum distance profile. Advances in Mathematics of Communications, 2016, 10 (2) : 275-290. doi: 10.3934/amc.2016005
[16]	Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028
[17]	Dean Crnković, Sanja Rukavina, Andrea Švob. Self-orthogonal codes from equitable partitions of distance-regular graphs. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022014
[18]	Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195
[19]	Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[20]	Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
José Joaquín Bernal Diana H. Bueno-Carreño Juan Jacobo Simón
on Google Scholar
José Joaquín Bernal Diana H. Bueno-Carreño Juan Jacobo Simón

[Back to Top]