Maximal AMDS codes. Complete (n,k)-arcs in PG(k-1,q) and projective (n,k) q -AMDS codes that admit no projective extensions are equivalent objects. We show that projective AMDS codes of reasonable length admit only linear extensions. Thus, we are able to prove the maximality of many known linear AMDS codes. At the same time our results sharply limit the possibilities for constructing long nonlinear AMDS codes. We also show that certain short linear AMDS codes are maximal. Central to our approach is the Bruen-Silverman model of linear codes first introduced in T. L. Alderson [On MDS codes and Bruen-Silverman codes. Ph.D. Thesis, University of Western Ontario, 2002)] and T. L. Alderson, A. A. Bruen, and R. Silverman [J. Combin. Theory Ser. A 114 (6), 1101–1117 (2007; Zbl 1126.94017)]