On additive mds codes over small fields

Official URL: https://dx.doi.org/10.3934/amc.2021024

Let C be a (n, q2k, 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 Fq2 then C is linear over Fq2. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over Fq for q ∈ {4, 8, 9}. We also classify the longest additive MDS codes over F16 which are linear over F4 . 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}.