American Institute of Mathematical Sciences

Advanced
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

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. Google Scholar

[2]

P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 963-1063.  Google Scholar

[3]

R. T. Chien and D. M. Choy, Algebraic generalization of BCH-Goppa-Helgert codes, IEEE Trans. Inf. Theory, 21 (1975), 70-79.  Google Scholar

[4]

, GAP - Groups, Algorithms, Programming - a system for computational discrete algebra,, , ().   Google Scholar

[5]

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

[6]

F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. Camion, Abelian codes, MRC Tech. Sum. Rep. 1059, Univ. Wisconsin Madison, 1970. Google Scholar

[2]

P. Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, North-Holland, Amsterdam, 1998, 963-1063.  Google Scholar

[3]

R. T. Chien and D. M. Choy, Algebraic generalization of BCH-Goppa-Helgert codes, IEEE Trans. Inf. Theory, 21 (1975), 70-79.  Google Scholar

[4]

, GAP - Groups, Algorithms, Programming - a system for computational discrete algebra,, , ().   Google Scholar

[5]

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

[6]

F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977. Google Scholar

[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.  Google Scholar

[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]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[20]

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

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (117)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences