Complete \(k\)-arcs in PG(\(n,q\)), \(q\) even
type: Item
instance of: scholarly article
title: Complete \(k\)-arcs in PG(\(n,q\)), \(q\) even
zbMATH Open document ID: 0759.51008
DOI: 10.1016/0012-365X(92)90576-2
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: \(PG(n,q)\) denotes the \(n\)-dimensional projective space over the field \(GF(q)\). A \(k\)-arc of points in it with \(k\geq n+1\) is a set \(K\) of \(k\) points with the property that no \(n+1\) points of \(K\) lie in a hyperplane. It may be noted that the study of \(k\)-arcs in \(PG(n,q)\) is interesting for coding theory also -- the \(k\)-arcs of \(PG(n,q)\) and linear MDS codes of dimension \(n+1\) and length over \(GF(q)\) are equivalent objects. In this paper, the authors investigate the completeness of \(k\)-arcs in \(PG(n,q)\) where \(q\) is even. All the values of \(k\) are determined, for which there exists a complete \(k\)-arc in \(PG(n,q)\), \(q-2\geq n > q- \sqrt{q}/2-11/4\). This is proved by using the duality principle between \(k\)-arcs in \(PG(n,q)\) and dual \(k\)-arcs in \(PG(k-n-2,q)\), \((k\geq n+4)\). The theorems show that the classification of all complete \(k\)-arcs in \(PG(n,q)\), \(q\) even at \(q-2\geq n > q-\sqrt{q}/2-11/4\) is closely related to the classification of all (q+2)-arcs in \(PG(2,q)\).
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 94553
zbMATH Keywords: \(k\)-arc
zbMATH Keywords: codes
zbMATH Keywords: PG(n,q)
zbMATH Keywords: complete arcs
author: Leo Storme
author: Joseph A. Thas
reviewed by: T. Thrivikraman
MaRDI profile type: Publication
cites work: Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre - some extensions
cites work: On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre
cites work: The solution to Beniamino Segre's problem \(I_ p,\),p=3, q=\(2^ k\).
cites work: On the uniqueness of \((q+1)_ 4\)-arcs of \(PG(4,q),q=2^ h,h\vee\)
cites work: Q3851217
cites work: Q3694152
cites work: Q4003762
cites work: Q4146667
cites work: Q3272923
cites work: M.D.S. codes and arcs in \(\mathrm{PG}(n,q)\) with \(q\) even: an improvement of the bounds of Bruen, Thas, and Blokhuis
cites work: \(k\)-arcs and dual \(k\)-arcs
cites work: Q5582819
cites work: Q5640331
cites work: Complete arcs and algebraic curves in \(\text{PG}(2,q)\)
cites work: Arcs in projective planes over prime fields
cites work: Q3361483
description: scientific article; zbMATH DE number 94553