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 \geq q+k and k+1 of the projections are F}_{q^2} then use this geometrical theorem, other geometric arguments some computations to classify all codes for q \in \{4,8,9\} also longest F}_{16} F}_4 In these cases, classifications not only verify conjecture codes, but confirm there no non-linear perform as well their counterparts. These results imply quantum holds \{ 2,3\}