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The classical example of an MDS code is the Reed-Solomon code, which is the evaluation code of all polynomials of degree at most k − 1 over Fq h . The Reed-Solomon code is linear over Fq h and has length q h + 1. The MDS conjecture states (excepting two specific cases) that an MDS code has length at most q h + 1. In other words, there are no better MDS codes than the Reed-Solomon codes. We use geometrical and computational techniques to classify all additive MDS codes over Fq h for q h ∈ {4, 8, 9}. We also classify the longest additive MDS codes over F16 which are linear over F4. These classifications not only verify the MDS conjecture for additive codes in these cases, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. In this talk I will cover the main geometrical theorem that allows us to obtain this classification and compare these classifications with the classifications of all MDS codes of alphabets of size at most 8, obtained previously by Alderson (2006), Kokkala, Krotov and Osterg˚ard (2015) and Kokkala  and Ostergard (2016)