On additive MDS codes over small fields

Abstract

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 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\} $.