Orthogonal geometries over finite fields with characteristic \(\neq 2\) and block designs (Q1314938)

Let \(q\) be an odd prime power and \(F_ p\) the finite field with \(q\) elements. Denote by \(\text{OV}_ n(F_ p)\) the \(n\)-dimensional orthogonal geometry over \(F_ p\). Based on the conjugation relationship between the subspaces in \(\text{OV}_ n(F_ p)\) which has been investigated by \textit{B. Yang} and \textit{W. Wei} in [Linear Algebra Appl. 134, 1-23 (1990; Zbl 0697.05015)], the authors take as blocks some subspace pairs and then construct some types of BIB designs (Theorems 1 and 2) and PBIB designs (Theorem 3 and 4 for two associate classes; Theorems 5 to 8 for more associate classes). However, they have a big value of parameters.