Coprimitive sets and inextendable codes

Complete (n, r)-arcs in PG(k − 1, q) and projective (n, k, n − r)q-codes that admit no projective extensions are equivalent objects. We show that projective codes of reasonable length admit only projective extensions. Thus, we are able to prove the maximality of many known linear codes. At the same time our results sharply limit the possibilities for constructing long nonlinear codes. We also show that certain short linear codes are maximal. The methods here may be just as interesting as the results. They are based on the Bruen-Silverman model of linear codes (see [1, 2, 9] and [15]) as well as the theory of Rédei blocking sets first introduced in [8].