Classical arcs in PG\((r,q)\) for \(23\leq q\leq 29\)
type: Item
instance of: scholarly article
title: Classical arcs in PG\((r,q)\) for \(23\leq q\leq 29\)
zbMATH Open document ID: 0977.51002
DOI: 10.1016/S0012-365X(00)00169-2
published in: Discrete Mathematics
publication date: 2002-01-21T00:00:00Z
review text: Let \(PG(r,q)\) be the \(r\)-dimensional projective space over Galois field \(F_q\). A \(k\)-arc \(K\) in \(PG(r,q)\) is a set of \(k\) points such that no \(r+1\) points of \(K\) lie in a hyperplane. A normal rational curve in \(PG(r,q)\) \((r\leq q-2)\) is a set of \(q+1\) points which is (projectively) isomorphic to \(\{(1,t, \dots,t^r)\); \(t\in F_q\cup \{\infty\}\}\). An arc is called classical or rational if it lies on a normal rational curve. Let \(m(r,q)\) be the maximum size of arcs in \(PG(r,q)\). The authors are interested in modifying B. Serge's conjecture on arcs: Let \(1\leq r\leq q-2\). (1) If \(q\) is odd, then \(m(r,q)=q+1\) and every \(m(r,q)\)-arc in \(PG(r,q)\) is classical. (2) If \(q\) is even, then \(m(r,q)=q+1\) except for \(m(2,q)=m (q-2,q)=q+2\), and every \(m(r,q)\)-arc in \(PG (r,q)\) is classical provided \(4\leq r\leq q-5\). The main results of this article, together with results for \(11\leq q\leq 19\) [see \textit{J. M. Chao} and \textit{H. Kaneta}, Discrete Math. 174, No. 1-3, 87-94 (1997; Zbl 0892.51007)] imply the following: Serge's conjecture holds for \(11\leq q\leq 27\). The main results are computational. All programmes are written in the language C.
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 1565956
zbMATH Keywords: classical arcs
zbMATH Keywords: Serge's conjecture
author: Jinming Chao
author: Hitoshi Kaneta
reviewed by: Pavel Burda
MaRDI profile type: Publication
full work available at URL: s0012-365x(00)00169-2
OpenAlex ID: W2916786102
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description: scientific article; zbMATH DE number 1565956