AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 55 (2013), Pages 205–233 Group developed weighing matrices ∗ K. T. Arasu Department of Mathematics & Statistics Wright State University 3640 Colonel Glenn Highway, Dayton, OH 45435 U.S.A. Jeffrey R. Hollon Department of Mathematics Sinclair Community College 444 W 3rd Street, Dayton, OH 45402 U.S.A. Abstract A weighing matrix is a square matrix whose entries are 1, 0 or -1, such that the matrix times its transpose is some integer multiple of the identity matrix. We examine the case where these matrices are said to be devel- oped by an abelian group. Through a combination of extending previous results and by giving explicit constructions we will answer the question of existence for 318 such matrices of order and weight both below 100. At the end, we are left with 98 open cases out of a possible 1,022. Further, some of the new results provide insight into the existence of matrices with larger weights and orders. 1 Introduction 1.1 Group Developed Weighing Matrices A weighing matrix W = W (n, k) is a square matrix, of order n, whose entries are in the set w i,j ∈ {-1, 0, +1}. This matrix satisfies WW t = kI n , where t denotes the matrix transpose, k is a positive integer known as the weight, and I n is the identity matrix of size n. Definition 1.1. Let G be a group of order n. An n × n matrix A =(a gh ) indexed by the elements of the group G (such that g and h belong to G) is said to be G-developed if it satisfies the condition a gh = a g+k,h+k for all g, h, k ∈ G. * Research partially supported by grants from the AFSOR and NSF.