A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices

A genetic algorithm for constructing cocyclic Hadamard matrices over a given group is described. The novelty of this algorithm is the guided heuristic procedure for reproduction, instead of the classical crossover and mutation operators. We include some runs of the algorithm for dihedral groups, which are known to give rise to a large amount of cocyclic Hadamard matrices.

[1]  Ilias S. Kotsireas,et al.  A meta-software system for the discovery of Hadamard matrices , 2005, 19th International Symposium on High Performance Computing Systems and Applications (HPCS'05).

[2]  K. Horadam.  Hadamard Matrices and Their Applications , 2006 .

[3]  D. Flannery,et al.  Cocyclic Hadamard Matrices and Hadamard Groups Are Equivalent , 1997 .

[4]  Andrés Iglesias,et al.  Mathematical Software - ICMS 2006, Second International Congress on Mathematical Software, Castro Urdiales, Spain, September 1-3, 2006, Proceedings , 2006, ICMS.

[5]  Shu Lin,et al.  Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , 1999, Lecture Notes in Computer Science.

[6]  K. J. Horadam,et al.  Generation of Cocyclic Hadamard Matrices , 1995 .

[7]  Víctor Álvarez,et al.  Calculating Cocyclic Hadamard Matrices in Mathematica: Exhaustive and Heuristic Searches , 2006, ICMS.

[8]  D. Flannery,et al.  Calculation of cocyclic matrices , 1996 .

[9]  Víctor Álvarez,et al.  A system of equations for describing cocyclic Hadamard matrices , 2008 .

[10]  Víctor Álvarez,et al.  The homological reduction method for computing cocyclic Hadamard matrices , 2009, J. Symb. Comput..

[11]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[12]  W. Wallis,et al.  Hadamard Matrices and Their Applications , 1978 .

[13]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[14]  V. Álvarez,et al.  Rooted Trees Searching for Cocyclic Hadamard Matrices over D4t , 2009, AAECC.

[15]  Víctor Álvarez,et al.  A Genetic Algorithm for Cocyclic Hadamard Matrices , 2006, AAECC.

[16]  Wieb Bosma,et al.  Computational Algebra and Number Theory , 1995 .

[17]  K. J. Horadam,et al.  Cocyclic Development of Designs , 1993 .

[18]  A. Baliga,et al.  Self-dual Codes Using Image Restoration Techniques , 2001, AAECC.