Inseparable Orthogonal Matrices over Z2

April 23, 1999

Abstract

(0, 1)-matrix A is called orthogonal over Z2 if both AAT and ATA are diagonal matrices. A matrix A is called inseparable if A contains no zero row nor zero column, and there do not exist permutation matrices P andQ such that

A matrix A is said to be of type 0 if AAT = 0 and ATA = 0. A square matrix A of order n is said to be of type 1 if AAT = In. It turns out that an inseparable orthogonal mtrix over Z2 is either of type 0 or of type 1.

A weighing matrix W (over the reals), of weight k and order n, is an n × n (0, 1, −1)-matrix satisfying WWT = kIn. Clearly, inseparable orthogonal matrices over Z2 are generalizations of weighing matrices. In this talk, we will briefly survey the known results and applications concerning weighting matrices, and then discuss the spectrum of the number of 1's in an inseparable orthogonal matrix over Z2. Eqquivalent formultion of the problem using graphs and its applications wil be given.