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

• 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.
Mathematics Subject Classification: 94B15, 94B65.

 Citation:

•  [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.

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