Complete spans on Hermitian varieties
type: Item
instance of: scholarly article
title: Complete spans on Hermitian varieties
zbMATH Open document ID: 1042.51002
DOI: 10.1023/A:1024179703511
author: Angela Aguglia
author: Antonio Cossidente
author: Gary L. Ebert
published in: Designs, Codes and Cryptography
publication date: 2003-08-21T00:00:00Z
review text: Consider the projective geometry \(\text{PG}(3,q)\), where \(q\) is a prime power, and let \({\mathcal L}\) be a general line complex in \(\text{PG}(3,q)\). The usual extension of scalars from \(\text{GF}(q)\) to \(\text{GF}(q^2)\) leads to an extension of the lines of \({\mathcal L}\) to lines of \(\text{PG}(3,q^2)\). The authors show these extended lines cover a nonsingular Hermitian surface \(H\cong H(3,q^2)\) in \(\text{PG}(3,q^2)\). If this extension is applied to a symplectic spread \({\mathcal S}\) of \(\text{PG}(3,q)\), the extended lines of \({\mathcal S}\) form a complete \((q^2+ 1)\)-span of \(H\). Using the software package MAGMA the authors generate such spans for small values of \(a\). These examples indicate that the known bounds (upper and lower) for the size of a complete span of \(H(3,q^2)\) are weak. They conjecture that good bounds are probably quadratic in \(q\). In the last part of this article a generalization of the above process to the Hermitian variety \(H(5,q^2)\) is given. Since spreads do not exist in dimension 5, the authors start with a symplectic subgeometry \(W_5(q)\) of the point set of \(H(5,q^2)\). This subgeometry has a spread and extending scalars from \(\text{GF}(q)\) to \(\text{GF}(q^2)\) gives a complete \((q^3+1)\)-span of \(H(5,q^2)\).
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1968852
zbMATH Keywords: Hermitian variety
zbMATH Keywords: complete partial spread
reviewed by: Michael J. Kallaher
describes a project that uses: Magma
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1968852