On additive MDS codes over small fields

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