On additive MDS codes over small fields

Let $C$ be a $(n,q^{2k},n-k+1{)}_{q^2}$ additive MDS code which is linear over ${\mathbb{F}}_q$. We prove that if $n⩾q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb{F}}_{q^2}$ then $C$ is linear over ${\mathbb{F}}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb{F}}_q$ for $q\in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb{F}}_{16}$ which are linear over ${\mathbb{F}}_4$. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $q\in \{2,3\}$.