On affine spaces of rectangular matrices with constant rank

Let $F$ be a field, and $n\ge p\ge r>0$ be integers. In a recent article, Rubei has determined, when $F$ is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in $F$ 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 $M_s\left(F\right)$ with dimension $\left(\begin{array}{c}s\\ 2\end{array}\right)$. The latter is known to be connected to the classification of nonisotropic quadratic forms over $F$ up to congruence.