On configurations of type nK with constant degree of irreducibility

Abstract

Symmetric schematic configurations C of type n κ correspond to n-square {0, 1}-matrices I which have κ entries 1 in each row and column, but no 2-square submatrices whose entries are all 1. With each anti-flag in C , i.e., with each entry 0 in I, we associate a weight which is defined as the number of certain 2-square submatrices of I. This paper deals with matrices I whose entries 0 have constant weight d. One has d = κ and d = 1 if, and only if, C is a projective plane of order κ − 1 and a generalized quadrangle or order (κ − 1, κ − 1), respectively. Finally, the addition and multiplication tables of GF( q) give rise to a third class of instances with parameters n = q 2, κ = q, and d = κ − 1 = q − 1.