Maximum distance separable codes and arcs in projective spaces

Abstract

Given any linear code C over a finite field GF ( q ) we show how C can be described in a transparent and geometrical way by using the associated Bruen–Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. [18]] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060–1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in PG ( n , q ) , MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1–3) (1990) 1–11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in PG ( n , q ) with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441–459].