Constructing a Canonical form of a Matrix in Several Problems about Combinatorial Designs

Abstract

The author developed computer programs needed for the classification of designs with certain automorphisms by the local approach method. All these programs use canonicity test or/and construction of canonical form of an integer matrix. Their efficiency substantially influences the speed of the whole computation. The present paper deals with the implemented canonicity algorithm. It is based on ideas used by McKay, Meringer, Kaski and Bouyukliev, but while their algorithms are for the equivalence test, the canonicity test or finding canonical representative of only one type of combinatorial object (graph, code, design, binary matrix, etc.), the algorithm presented in this paper is meant to work fast on all types of integer matrices used for the classification of designs with predefined automorphisms. This is achieved through the suitable spectrum invariant, and the way it is used to cut off some branches of the search tree.