Lattices Associated with Finite Vector Spaces and Finite Affine Spaces

Abstract

Let \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) be the \(n\)-dimensional vector (resp. affine) space over the finite field \(\mathbb{F}_q\). For \(1 \leq i \leq i+s \leq n-1\) (resp. \(0 \leq i \leq i+s \leq n-1\)), let \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) denote the set of all subspaces (resp. flats) in \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) with dimensions between \(i\) and \(i+s\) including \(\mathbb{F}_q^(n)\) and \(\{0\}\) (resp. \(\emptyset\)). By ordering \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) by ordinary inclusion or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity.