Abstract
We describe a notebook in Mathematica which, taking as input data a homological model for a finite group G of order |G| = 4t, performs an exhaustive search for constructing the whole set of cocyclic Hadamard matrices over G. Since such an exhaustive search is not practical for orders 4t ≥28, the program also provides an alternate method, in which an heuristic search (in terms of a genetic algorithm) is performed. We include some executions and examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Álvarez, V.: http://mathworld.wolfram.com/HadamardSearch.html (to appear, 2006)
Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: An algorithm for computing cocyclic matrices developed over some semidirect products. In: Bozta, S., Sphparlinski, I. (eds.) AAECC 2001. LNCS, vol. 2227, pp. 287–296. Springer, Heidelberg (2001)
Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: A genetic algorithm for cocyclic Hadamard matrices. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 2006. LNCS, vol. 3857, pp. 144–153. Springer, Heidelberg (2006)
Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: Homological reduction method for constructing Hadamard cocyclic matrices. In: Communication submitted to the EACA 2006 conference, Sevilla (2006)
Baliga, A., Chua, J.: Self-dual codes using image resoration techniques. In: Bozta, S., Sphparlinski, I. (eds.) AAECC 2001. LNCS, vol. 2227, p. 46. Springer, Heidelberg (2001)
Baliga, A., Horadam, K.J.: Cocyclic Hadamard matrices over \(Z_t X Z_2^2\). Australas. J. Combin. 11, 123–134 (1995)
Dousson, X., Rubio, J., Sergeraert, F., Siret, Y.: The Kenzo program. Institute Fourier, Grenoble (1998), http://www-fourier.ujf-grenoble.fr/~sergeraert/Kenzo/
de Launey, W., Horadam, K.J.: Cocyclic development of designs. J. Algebraic Combin. 2(3), 129 (1993)
de Launey, W., Horadam, K.J.: Generation of cocyclic Hadamard matrices. In: Computational algebra and number theory, Sydney. Math. Appl, vol. 325, pp. 279–290. Kluwer Acad. Publ, Dordrecht (1995)
Flannery, D.L.: Calculation of cocyclic matrices. J. of Pure and Applied Algebra 112, 181–190 (1996)
Flannery, D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192, 749–779 (1997)
Flannery, D.L., O’Brien, E.A.: Computing 2-cocycles for central extensions and relative difference sets. Comm. Algebra. 28(4), 1939–1955 (2000)
The GAP group, GAP- Group, Algorithms and programming, School of Mathematical and Computational Sciences. University of St. Andrews, Scotland (1998)
Grabmeier, J., Lambe, L.A.: Computing Resolutions Over Finite p-Groups. In: Betten, A., Kohnert, A., Lave, R., Wassermann, A. (eds.) Proceedings ALCOMA 1999. Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2000)
Ellis, G.: GAP package HAP, Homological Algebra Programming, http://hamilton.nuigalway.ie/Hap/www/
Horadam, K.: Progress in cocyclic matrices. Congressus numerantium 118, 161–171 (1996)
The MAGMA computational algebra system, http://magma.maths.usyd.edu.au
Mac Lane, S.: Homology. Classics in Mathematics. Springer, Berlin (1995)(Reprint of the 1975 edition)
Veblen, O.: Analisis situs, vol. 5. A.M.S. Publications (1931)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Álvarez, V., Armario, J.A., Frau, M.D., Real, P. (2006). Calculating Cocyclic Hadamard Matrices in Mathematica: Exhaustive and Heuristic Searches. In: Iglesias, A., Takayama, N. (eds) Mathematical Software - ICMS 2006. ICMS 2006. Lecture Notes in Computer Science, vol 4151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11832225_42
Download citation
DOI: https://doi.org/10.1007/11832225_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38084-9
Online ISBN: 978-3-540-38086-3
eBook Packages: Computer ScienceComputer Science (R0)