Abstract
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 non-linear 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 Alderson TL (2002) PhD. Thesis, University of Western Ontario; Alderson TL (to appear) J Combin Theory Ser A; Bruen AA, Silverman R (1988) Geom Dedicata 28(1): 31---43; Silverman R (1960) Can J Math 12: 158---176) as well as the theory of Redei blocking sets first introduced in Bruen AA, Levinger B (1973) Can J Math 25: 1060---1065.
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