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

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 \(UV_n({F}_{q^2})\) 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 \(UV_n({F}_{q^2})\), acts transitively on some subsets of subspaces of \(UV_n({F}_{q^2})\). This observation gives rise to a number of Partially Balanced Incomplete Block Designs (PBIBD’s).