Combinatorics of Euclidean Spaces over Finite Fields

The q-binomial coefficients are q-analogues of the binomial coefficients, counting the number of k-dimensional subspaces in the n-dimensional vector space \({\mathbb {F}}^n_q\) over \({\mathbb {F}}_{q}.\) In this paper, we define a Euclidean analogue of q-binomial coefficients as the number of k-dimensional subspaces which have an orthonormal basis in the quadratic space \(({\mathbb {F}}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}).\) We prove its various combinatorial properties compared with those of q-binomial coefficients. In addition, we formulate the number of subspaces of other quadratic types and study some related properties.