Searching for partial Hadamard matrices

Details Searching for partial Hadamard matrices Searching for partial Hadamard matrices

2012-01-19

arxiv.org/abs/1201.4021v1

Mathematics

Combinatorics

23 pages, 2 figures

Scientific paper

Three algorithms looking for pretty large partial Hadamard matrices are described. Here "large" means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [dLa00]) is achieved. The first one performs some kind of local exhaustive search, and consequently is expensive from the time consuming point of view. The second one comes from the adaptation of the best genetic algorithm known so far searching for cliques in a graph, due to Singh and Gupta [SG06]. The last one consists in another heuristic search, which prioritizes the required processing time better than the final size of the partial Hadamard matrix to be obtained. In all cases, the key idea is characterizing the adjacency properties of vertices in a particular subgraph G_t of Ito's Hadamard Graph Delta (4t) [Ito85], since cliques of order m in G_t can be seen as (m+3)*4t partial Hadamard matrices.

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