Completeness of normal rational curves
type: Item
instance of: scholarly article
title: Completeness of normal rational curves
zbMATH Open document ID: 0771.51005
DOI: 10.1023/A:1022428405084
published in: Journal of Algebraic Combinatorics
publication date: 1993-03-15T00:00:00Z
review text: A \(k\)-arc in \(\mathrm{PG}(n,q)\), the \(n\)-dimensional projective space over \(\mathrm{GF}(q)\), is a set \(K\) of \(k\) points of \(\mathrm{PG}(n,q)\), \(k\geq n+1\), with the property that no \(n+1\) points of \(K\) lie in a hyperplane. A normal rational curve in \(\mathrm{PG}(n,q)\), \(2\leq n\leq q-2\), is a \((q+1)\)-arc that is projectively equivalent to the \((q+1)\)-arc given, in homogeneous coordinates, by \(\{(1,t,\dots,t^ n)\); \(t\in \mathrm{GF}(q)\}\cup\{(0,\dots,1)\}\). In \(\mathrm{PG}(q-1,q)\), all \((q+1)\)-arcs are called normal rational curves. The author investigates whether a normal rational curve of \(\mathrm{PG}(n,q)\) can be extended to a \((q+2)\)-arc. He improves previous results of Storme and Thas, using a result by Kovács. The problem is now completely solved for large prime numbers \(q\) and odd nonsquare prime powers \(q=p^{2h+1}\) with \(p\) prime, \(p\ge p_0(h)\), \(h\ge 1\), where \(p_0(h)\) is an odd prime number which depends on \(h\).
reviewed by: Luc Teirlinck
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 130430
zbMATH Keywords: \(k\)-arc
zbMATH Keywords: normal rational curve
zbMATH Keywords: MDS code
author: Leo Storme
MaRDI profile type: Publication
cites work: Q3694152
cites work: An elementary proof and an extension of Thas' theorem on k-arcs
cites work: Q4018885
cites work: On MDS extensions of generalized Reed- Solomon codes
cites work: Q3357797
cites work: \(k\)-arcs and dual \(k\)-arcs
cites work: Q5582819
cites work: Q5640331
cites work: Q4764143
cites work: Arcs in projective planes over prime fields
cites work: Q3361483
description: scientific article; zbMATH DE number 130430