Quasi-semisymmetric designs with extremal conditions

Abstract

A finite incidence structure Pi = (X,B) is called a quasi-semisymmetric design (QSSD) with nexus alpha if there exist positive integers lambda, mu, and alpha such that any two distinct points are in 0 or lambda common blocks, any two distinct blocks are incident with 0 or mu common points, and for each nonincident point-block pair (x,B), there are exactly alpha blocks B' with x is an element of B' and B' boolean AND B not equal theta. Symmetric designs, semisymmetric designs, and partial lambda-geometries are among such structures. In this paper, in addition to some general properties, we study the existence conditions for QSSDs with mu = lambda - 1 greater than or equal to 2 and the properties of QSSDs satisfying the following extremal condition: if B-1 and B-2 are two blocks with a nonempty intersection, then there are another lambda - 2 blocks B-3,...,B-lambda such that boolean AND(1) less than or equal to i less than or equal to lambda B-i = B-1 boolean AND B-2. We show that alpha greater than or equal to (lambda(2)(mu-1)+lambda)/mu under such a condition, and QSSDs with equality are classified whenever mu = lambda or mu = lambda - 1 following a classification of affine polar spaces by Cohen and Shult (Geometraic Dedicata 35 (1990), 43-76).