Generating binary partial Hadamard matrices

Abstract

This paper deals with partial binary Hadamard matrices. Although there is a fast simple way to generate about a half (which is the best asymptotic bound known so far, see de Launey (2000) and de Launey and Gordon (2001)) of a full Hadamard matrix, it cannot provide larger partial Hadamard matrices beyond this bound. In order to overcome such a limitation, we introduce a particular subgraph Gt of Ito’s Hadamard Graph Δ(4t) (Ito, 1985), and study some of its properties,which facilitates that a procedure may be designed for constructing large partial Hadamard matrices. The key idea is translating the problem of extending a given clique in Gt into a Constraint Satisfaction Problem, to be solved by Minion (Gent et al., 2006). Actually, iteration of this process ends with large partial Hadamard matrices, usually beyond the bound of half a full Hadamard matrix, at least as our computation capabilities have led us thus far.