# American Institute of Mathematical Sciences

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

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

 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.
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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