Deep holes of full-length Reed--Solomon codes: uniform bounds and exact classifications

Let $u$ be a received word for the full-length Reed--Solomon code $\operatorname{RS}_q(k)$, with interpolation polynomial $f_u$ of degree $k+m$. For each $m\in\{3,4,5,6,7\}$, we classify all such deep holes over every finite field, without a field-size hypothesis. None occur for odd $m$ in this rang