On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, q) and Extendability of Reed-Solomon Codes

A subset S of a conic C in the projective plane PG(2, q) is called almost complete (AC-subset for short) if it can be extended to a larger arc in PG(2, q) only by the points of C S and by the nucleus of C when q is even. New upper bounds on the smallest size t(q) of an AC-subset are obtained. The new bounds are used to increase regions of pairs (N, q) for which it is proved that every normal rational curve in PG(N, q) is a complete (q + 1)-arc or, equivalently, that none [q +1,N +1, q −N +1]q (resp. [q+1, q −N,N +2]q) generalized doubly- extended Reed-Solomon code cannot be extended to a [q + 2,N + 1, q − N + 2]q (resp. [q + 2, q − N + 1,N + 2]q) MDS code.