Finite Unitary Geometries and PBIBD’s $(II)$

Abstract

Let $q$ be a prime power, $F_{q^2}$ the finite field with $q^2$ elements, $U_n(F_{q^2})$ the finite unitary group of degree $n$ over $F_{q^2}$, and $U{V}_n(F_{q^2})$ the $n$-dimensional unitary geometry over $F_{q^2}$. It is proven that the subgroup consisting of the elements of $U_n(F_{q^2})$ which fix a given $(m,s)$-type subspace of $U{V}_n(F_{q^2})$, acts transitively on some subsets of subspaces of $U{V}_n(F_{q^2})$. This observation gives rise to a number of Partially Balanced Incomplete Block Designs (PBIBD’s).