On cyclic caps in 4-dimensional projective spaces
type: Item
instance of: scholarly article
title: On cyclic caps in 4-dimensional projective spaces
zbMATH Open document ID: 1196.51009
published in: Designs, Codes and Cryptography
publication date: 2009-03-31T00:00:00Z
review text: A cap in a projective space \(\text{PG}(N,q)\) is a set of points no three of which are collinear; a cap is called complete if it is not contained in a larger cap. A cap is said to be cyclic if it is an orbit under a cyclic group of collineations of \(\text{PG}(N,q)\). The complement of a hyperplane in \(\text{PG}(N,q)\) can be naturally identified with the field \(\mathbb F_{q^ N}\) in such a way that the multiplicative group of \(\mathbb F_{q^ N}\) acts as a cyclic collineation group of \(\text{PG}(N,q)\). For \(N=4\), it is known that under this identification the multiplicative group \(C_{q^ 2 +1}\) of order \(q^ 2 +1\) is a cyclic cap. The author shows that this cap is never complete. If \(q\) is odd, \(m \geq 2\) and \(C_{m(q^ 2 +1)}\) is a cap, then \(m =2\) and \(q \equiv 3 \pmod 4\). Moreover, the resulting cap \(C_{2(q^ 2 +1)}\) is complete.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 5536724
zbMATH Keywords: cyclic cap
zbMATH Keywords: complete cap
zbMATH Keywords: BCH-code
zbMATH Keywords: quasi-perfect code
author: Massimo Giulietti
MaRDI profile type: Publication
full work available at URL: s10623-007-9106-1
OpenAlex ID: W1978646811
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DOI: 10.1007/S10623-007-9106-1
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description: scientific article; zbMATH DE number 5536724