Orthogonal designs III: Weighing matrices

Abstract

A weighing matrix W = W(n,k) of order n and weight k is a square (0,l,-l)-matrix satisfying\nWWt -kIn\nAn 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.\nThis paper reviews and updates the current status of the conjectures:\nI. 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;\nII. Let n = 0 (mod 4). Then there exists a W(n,k) for each k < n. This conjecture has been verified for n = 28, 2t+l, 2t +l·3 and 2t+1.5, where t is any positive integer;\nIII. 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;\nIV. Let n = 0 (mod 8). Then there exists an orthogonal design (1,k) for all k < n. . This conjecture has been verified for n = 2t+2, 2t+2.3 and 2t+2.5, where t is any positive integer.\nV. 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.