On affine spaces of rectangular matrices with constant rank

Let $\mathbb{F}$ be a field, and $n\ge p\ge r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$–by–$p$ matrices with entries in $\mathbb{F}$ in which all the elements have rank $r$. In this note, we generalize her result to an arbitrary field with more than $r+1$ elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of ${\mathrm{M}}_s(\mathbb{F})$ with dimension ${\displaystyle \left(\genfrac{}{}{0px}{}{s}{2}\right)}$. The latter is known to be connected to the classification of nonisotropic quadratic forms over $\mathbb{F}$ up to congruence.