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D-optimal weighing designs for n -1 (mod 4) objects and a large number of weighings
 

Summary: D-optimal weighing designs for n -1 (mod 4) objects and a
large number of weighings
Bernardo M. ´Abrego
Silvia Fern´andez-Merchant
Michael G. Neubauer
William Watkins
California State University, Northridge
October 3, 2002
Abstract
Let Mm,n(0, 1) denote the set of all m × n (0,1)-matrices and let
G(m, n) = max{det XT
X : X Mm,n(0, 1)}.
In this paper we exhibit some new formulas for G(m, n) where n -1 (mod 4). Specifically, for m =
nt+r where 0 r < n, we show that for all sufficiently large t, G(nt+r, n) is a polynomial in t of degree
n that depends on the characteristic polynomial of the adjacency matrix of a certain regular graph. Thus
the problem of finding G(nt + r, n) for large t is equivalent to finding a regular graph, whose degree of
regularity and number of vertices depend only on n and r, with a certain "trace-minimal" property. In
particular we determine the appropriate trace-minimal graph and hence the formulas for G(nt +r, n) for
n = 11, 15, all r, and all sufficiently large t.
Keywords: D-optimal design, statistical design, regular graph

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics