Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes

We prove a stabilization result for the $\mathbb{F}_q$-dimension of spaces of morphisms between supersingular Drinfeld modules, filtered by degree: for any two supersingular rank-$2$ Drinfeld $\mathbb{F}_q[T]$-modules in characteristic $\frak{p}$ of degree $d$, the dimension $m_s$ of the space of morphisms of $\tau$-degree at most $s$ satisfies $m_s = 2(s+1)-(d-1)$ for all $s\geq d-2$. This is proved using the theory of Brandt matrices and properties of $L$-functions of automorphic forms for $\mathrm{GL}_2$ over function fields. The stabilization formula, combined with an analysis of zero entries in Brandt matrices and a hyperplane-avoidance argument, yields semifield rank-metric codes. We also describe an efficient algorithm for computing the relevant Brandt matrices.