ACS searching for D4t-Hadamard matrices

Authors:
Víctor Álvarez;José Andrés Armario;María Dolores Frau;Félix Gudiel;Belén Güemes;Elena Martín;Amparo Osuna
Affiliations:
University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain;University of Sevilla, Sevilla, Spain
Venue:
ANTS'10 Proceedings of the 7th international conference on Swarm intelligence
Year:
2010

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Abstract

An Ant Colony System (ACS) looking for cocyclic Hadamard matrices over dihedral groups D4t is described. The underlying weighted graph consists of the rooted trees described in [1], whose vertices are certain subsets of coboundaries. A branch of these trees defines a D4t- Hadamard matrix if and only if two conditions hold: (i) Ii = i - 1 and, (ii) ci = t, for every 2 ≤ i ≤ t, where Ii and ci denote the number of i- paths and i-intersections (see [3] for details) related to the coboundaries defining the branch. The pheromone and heuristic values of our ACS are defined in such a way that condition (i) is always satisfied, and condition (ii) is closely to be satisfied.