Inseparable Orthogonal Matrices over Z2

A (0, 1)-matrixAis called orthogonal over Z2if bothAATandATAare diagonal matrices. A matrixAis called inseparable ifAcontains no zero row or zero column and there do not exist permutation matricesPandQsuch that[formula]A matrixAis said to be of type 0 ifAAT=OandATA=O. A square matrixAof ordernis said to be of type 1 ifAAT=In. It turns out that an inseparable orthogonal matrix over Z2is either of type 0 or of type 1. Letf0(m,=n) (respectively,F0(m, n)) denote the smallest (respectively, largest) number of 1ʹs in anm×ninseparable orthogonal matrix of type 0 over Z2, andf1(n) (respectively,F1(n)) denote the smallest (respectively, largest) number of 1ʹs in ann×ninseparable orthogonal matrix of type 1 over Z2. The formulas forf0(m, n),F0(m, n),f1(n), andF1(n) are completely determined in this paper.