Applications of finite fields to combinatorics and finite geometries
type: Item
instance of: scholarly article
title: Applications of finite fields to combinatorics and finite geometries
zbMATH Open document ID: 1124.05017
author: Aiden A. Bruen
published in: Acta Applicandae Mathematicae
publication date: 2006-11-06T00:00:00Z
review text: This paper on applications of finite fields to combinatorics and finite geometry is in three parts. In the first part the author obtains a bound on the degree of a polynomial \(f\in F[x_1,\dots,x_n]\), \(F= \text{GF}(q)\) satisfying some local conditions with respect to its action on the points of the \(n\)-dimensional affine space AG\((n, F)\). The result yields an interesting application on a \(t\)-intersection set \(W\) in AG\((n, F)\), which is a set of points in \(\Omega= \text{AG}(n, F)\), such that each hyperplane in \(\Omega\) contains at least \(t> 0\) points of \(W\). By using the bound, such a subset of the affine space contains at least \((n+ t- 1)(q- 1)+1\) points. The second part of the paper is devoted to the analysis of the \(p\)-rank of an incidence matrix of a finite linear space with respect to the prime field GF\((p)\). The author gives an important lower bound of the \(p\)-rank of such a matrix. Especially, his result says that the \(p\)-rank of a finite projective plane is at least \(n\sqrt{n}+ 1\), where \(p\) is a prime dividing the order \(n\) of the plane. Finally, the last part deals with certain MDS codes and BRUCK nets. In particular, the author constructs a class of so-called unimbeddable nets.
reviewed by: Udo Ott
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 5070968
zbMATH Keywords: finite fields
zbMATH Keywords: affine space
zbMATH Keywords: MDS codes
MaRDI profile type: Publication
full work available at URL: s10440-006-9051-4
OpenAlex ID: W2013358658
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cites work: Covering finite fields with cosets of subspaces
DOI: 10.1007/S10440-006-9051-4
description: scientific article; zbMATH DE number 5070968