On Subsets of the Normal Rational Curve.

A normal rational curve of the (k-1) -dimensional projective space over {\mathbb F}_{q} is an arc of size q+1$ , since any k$ points of the curve span the whole space. In this paper, we will prove that if q$ is odd, then a subset of size 3k-6$ of a normal rational curve cannot be extended to an arc of size q+2$ . In fact, we prove something slightly stronger. Suppose that q$ is odd and E$ is a (2k-3)$ -subset of an arc G$ of size 3k-6$ . If G$ projects to a subset of a conic from every (k-3)$ -subset of E of odd characteristic, which can be extended to a Reed–Solomon code of length $q+1$ , cannot be extended to a linear maximum distance separable code of length $q+2$ . [ABSTRACT FROM AUTHOR]