On Subsets of the Normal Rational Curve

A normal rational curve of the (k - 1)-dimensional projective space over Fq 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, then G cannot be extended to an arc of size q + 2. Stated in terms of errorcorrecting codes we prove that a k-dimensional linear maximum distance separable code of length 3k - 6 over a field Fq 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.