2016/03/22 by Simeon Ball, Ball, Simeon, Jan De Beule +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1603.06714
openalex publication_date 2016/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A normal rational curve of the (k-1)-dimensional projective space over \mathbb Fq 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 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.