Mathematics – Combinatorics
Scientific paper
2011-12-20
Mathematics
Combinatorics
18 pages, 3 figures, 5 tables (figures corrected in v4). Presented at the International Workshop on Hadamard Matrices held in
Scientific paper
The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant H(n) of a square {+1, -1} matrix of given order n. Such a matrix with maximum determinant is called a D-optimal design of order n. We consider some cases where n is not divisible by 4, so the Hadamard bound is not attainable, but bounds due to Barba or Ehlich and Wojtas may be attainable. If R is a matrix with maximal (or conjectured maximal) determinant, then G = RR^T is the corresponding Gram matrix. For the cases that we consider, maximal or conjectured maximal Gram matrices are known. We show how to generate many Hadamard equivalence classes of solutions from a given Gram matrix G, using a randomised decomposition algorithm and row/column switching. In particular, we consider orders 26, 27 and 33, and obtain new D-optimal designs (for order 26) and new conjectured D-optimal designs (for orders 27 and 33).
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