Eduardo Piza^†

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Resumen

En este artírculo estudiamos las matrices de Hadamard y algunos algoritmos para generarlas. Revisamos varios aspectos teóricos en torno a la conjetura de Hadamard, que afirma que todo entero positivo múltiplo de 4 es un número de Hadamard. Posteriormente se describen los métodos de Kronecker, Sylvester, Paley, Williamson, Goethals-Seidel, Cooper-Wallis, Baumert-Hall, Ehlich y conjuntos diferencia suplementarios. Se establece la criba de Hadamard: 668 es el menor orden para el cual se desconoce si existe una matriz de Hadamard. Finalmente proponemos algoritmos de recocido simulado para hallar matrices de Hadamard a partir de secuencias Turyn. Hallamos excelentes soluciones con este método de búsqueda.

Palabras clave: matrices de Hadamard, recocido simulado, optimización combinatoria.

Abstract

In this paper we study the Hadamard matrices and some algorithms to generate them. We review some theoretical aspects about Hadamard's conjecture, which asserts that every positive integer multiple of 4 is a Hadamard number. Then we describe the methods of Kronecker, Sylvester, Paley, Williamson, Goethals-Seidel, Cooper-Wallis, Baumert-Hall, Ehlich and supplementary difference sets. Subsequently we settle the Hadamard sieve: 668 is lowest order for which is unknown if there exist an Hadamard matrix. Finally we propose a simulated annealing algorithms as alternative to find Hadamard matrices from Turyn sequences. We found excellent solutions with this search method.

Keywords: Hadamard matrices, simulated annealing, combinatorial optimization.

Mathematics Subject Classification: 15B34, 05B20, 90C27.

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Referencias

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Correspondencia a: Eduardo Piza. Centro de Investigación en Matemática Pura y Aplicada (CIMPA), Universidad de Costa Rica. San José, Costa Rica. E-Mail:  eduardojpiza@hotmail.com

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Received: 26 Aug 2010; Revised: 9 May 2011; Accepted: 10 May 2011