On subsets of the normal rational curve

Abstract

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 article 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 error-correcting codes we prove that a $k$-dimensional linear maximum distance separable code of length $3k-6$ over a field ${\mathbb{F}}_q$ 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 $q+2$.