Translation planes and derivation sets
type: Item
instance of: scholarly article
title: Translation planes and derivation sets
zbMATH Open document ID: 0705.51011
DOI: 10.1007/BF01230354
published in: Journal of Geometry
publication date: 1990-01-01T00:00:00Z
review text: The authors continue their work on coding theoretic techniques applied to finite geometry. Let \(\pi\) be an affine plane of order n, and let p be a prime dividing n. Let \(N=n^ 2\), \(F_ p\) the finite field of order p, and \(F^ N_ p\) the vector space of all functions from the point set of \(\pi\) to \(F_ p\). Let \(C_ p(\pi)\) be the subspace of \(F^ N_ p\) generated by the characteristic functions of the lines of \(\pi\). If \(\pi\) is a translation plane, then certainly \(n=q=p^ s\) for some prime p and some positive integer s. Let \(K=F_ q\) and view \(V=K\oplus K\) as a 2s dimensional vector space over \(F_ q\). Let \(F^ V_ p\) be the vector space of all functions from V to \(F_ p\). Let F be any subfield of K, and let B(K\(| F)\) be the subspace of \(F^ V_ p\) generated by all characteristic vectors of F-flats (s-dimensional flats of V that are cosets of F-subspaces of V). Say that \(\pi\) is ``contained'' in B(K\(| F)\) if and only if \(C_ p(\pi)\) is isomorphic to a subcode of B(K\(| F)\). This equivalent to saying \(\pi\) has kernel F in the usual geometric sense. Finally, let E(K\(| F)\) be the subspace of \(F^ V_ p\) generated by all differences of characteristic vectors \(v^ X-v^ Y\), where X and Y are parallel F-flats of V. The authors use this machinery to give a new upper bound for the p-rank of \(\pi\) ; namely, they show \(\dim (C_ p(\pi))\leq q+\dim (E(K| F))\) whenever \(\pi\) is a translation plane contained in B(K\(| F)\) as above. However, the main purpose of the paper is to give a general notion of derivation set for any finite projective plane by using coding theoretic ideas similar to those above. The ``classical'' derivation uses a Baer segment D on the line L at infinity and a collection of Baer subplanes having L as a line such that D is the intersection of L with each of these Baer subplanes. The authors show that for translation planes the ``classical'' Baer subplane derivation takes place in B(K\(| F)\) as defined above. Specializing…
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 94B27
zbMATH DE Number: 4157371
zbMATH Keywords: codes
zbMATH Keywords: translation plane
zbMATH Keywords: derivation set
zbMATH Keywords: finite projective plane
author: Edward F. jun. Assmus
author: J. D. Key
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01230354
OpenAlex ID: W2030139816
description: scientific article; zbMATH DE number 4157371