# 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 𝔽_q. We prove that if n ⩾ q+k and k+1 of the projections of C are linear over 𝔽_q^2 then C is linear over 𝔽_q^2. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over 𝔽_q for q ∈{4,8,9}. We also classify the longest additive MDS codes over 𝔽_16 which are linear over 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 ∈{ 2,3}.

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