Orthogonal designs III: Weighing matrices

Authors

Anthony V. Geramita
Jennifer Seberry, University of WollongongFollow

Publication Details

Anthony V. Geramita and Jennifer Seberry Wallis, Orthogonal designs III: Weighing matrices, Utilitas Mathematica, 6, (1974), 209-236.

Abstract

A weighing matrix W = W(n,k) of order n and weight k is a square (0,l,-l)-matrix satisfying

WW^t -kI_n

An orthogonal design of order n on a single variable is a weighing matrix and consequently the study of orthogonal designs is intimately connected with the study of weighing matrices.

This paper reviews and updates the current status of the conjectures:

I. Let n = 2 (mod 4). Then there exists a W(n,k) if and only if k < n - 1 is the sum of two integer squares;

II. Let n = 0 (mod 4). Then there exists a W(n,k) for each k < n. This conjecture has been verified for n = 28, 2^t+l, 2^{t +l}·3 and 2^t+1.5, where t is any positive integer;

III. Let n = 4 (mod 8). Then there exists an orthogonal design (1,1) for all k < n where k is the sum of three integer squares;

IV. Let n = 0 (mod 8). Then there exists an orthogonal design (1,k) for all k < n. . This conjecture has been verified for n = 2^t+2, 2^t+2.3 and 2^t+2.5, where t is any positive integer.

V. Let n = 2 (mod 4). Then there exists an orthogonal design of type (l,k) in order n for all k < n - 1 such that k = a², a an integer.