Finding D-optimal designs by randomised decomposition and switching
Abstract
The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant D(n) of a square {+1, -1} matrix of given order n. Such a matrix with maximum determinant is called a saturated D-optimal design. We consider some cases where n > 2 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 saturated D-optimal designs (for order 26) and new conjectured saturated D-optimal designs (for orders 27 and 33).
Cite
@article{arxiv.1112.4671,
title = {Finding D-optimal designs by randomised decomposition and switching},
author = {Richard P. Brent},
journal= {arXiv preprint arXiv:1112.4671},
year = {2014}
}
Comments
18 pages, 3 figures, 5 tables (figures corrected in v4). v5 added a reference and made minor improvements. Presented at the International Workshop on Hadamard Matrices held in honour of Kathy Horadam's 60th birthday, Melbourne, Nov. 2011. Data files are available at http://maths.anu.edu.au/~brent/maxdet/