Nonexistence Results for Hadamard-like Matrices

Abstract

The class of square $(0,1,-1)$-matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an $n$ by $n$ $(0,1,-1)$-matrix whose rows are nonzero, mutually orthogonal and whose first row has no zeros, then $n$ is not of the form $p^k$, $2p^k$ or $3p$ where $p$ is an odd prime, and $k$ is a positive integer.