ON ADDITIVE MDS CODES OVER SMALL FIELDS

Let C be a (n, q(2k), n - k + 1)(q2) additive MDS code which is linear over Fq. We prove that if n >= q + k and k + 1 of the projections of C are linear over F-q2 then C is linear over F-q2. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over Fq for q is an element of {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 is an element of {2, 3}.