An Isometric Invariant of Quadratic Spaces over Finite Fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field with an odd prime power $q$. In this paper, we construct a new isometric invariant of combinatorial type on $({\mathbb{F}}_q^n,{\text{dot}}_n)$, where ${\text{dot}}_n(\mathbf{x}):=x_1^2+\cdots +x_n^2$. Additionally, using counts from our new invariant, we give a new proof of Minkowski's formula on the size of spheres over finite fields. We also show which types of quadratic subspaces can be embedded in $({\mathbb{F}}_q^n,{\text{dot}}_n)$.

The results are already well-known

References in corpus (1)

• Embeddings of quadratic spaces over the field of $p$-adic numbers