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

José Joaquín  Bernal; Diana H.  Bueno-Carreño; Juan Jacobo  Simón

doi:10.3934/amc.2016018

Article Contents

2016, Volume 10, Issue 2: 459-474. Doi: 10.3934/amc.2016018

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

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

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

Received: September 30, 2014

Revised: August 31, 2015

Published: April 2016

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 94B15, 94B65.

Citation:

Related Papers

Cited by

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.