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

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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
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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,, , ().
[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.

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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,, , ().
[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.

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