Counting Configurations of Vectors in a Finite Vector Space with an Orthogonal, Symplectic or Unitary Geometry

Abstract

Given a finite-dimensional vector space $V$ over a finite field $F$ of odd characteristic, and equipping $V$ with an orthogonal (symplectic, unitary) geometry, the following two questions are considered:

1. Given some linearly independent vectors $w_1,w_2,\dots ,w_k\in V$ and the $k\times k$ matrix $A=(⟨w_i,w_j⟩)$, and given scalars ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k,\beta \in F$, how many vectors $v\in V$, not in the linear span of $w_1,w_2,\dots ,w_k$, satisfy $⟨w_i,v⟩={\alpha }_i$ ($i=1,2,\dots ,k$) and $⟨v,v⟩=\beta$?
2. Given a $k\times k$ matrix $A=({\lambda }_{ij})$ with entries from $F$, how many $k$-tuples $(v_1,v_2,\dots ,v_k)$ of linearly independent vectors from $V$ satisfy $⟨v_i,v_j⟩={\lambda }_{ij}$ ($i,j=1,2,\dots k$)?

An exact answer to the first question is derived. Here there are two cases to consider, depending on whether or not the column vector $({\alpha }_i)$ is in the column space of $A$. This result can then be applied iteratively to address the second question.