On the graph of a function in many variables over a finite field
type: Item
instance of: scholarly article
title: On the graph of a function in many variables over a finite field
zbMATH Open document ID: 1204.51009
author: Simeon Ball
published in: Designs, Codes and Cryptography
publication date: 2009-03-31T00:00:00Z
review text: The paper contains some improved bounds on the number of directions not determined by a point set in the affine space \(AG(k, q)\). Let \(x\) and \(y\) be two distinct points of \(AG(k, q)\), the \textit{direction} determined by \(x\) and \(y\) is the point \(d= \langle x - y \rangle = \{\lambda(x-y) \mid \lambda \in GF(q)\}\) of \(PG(k-1, q)\). The following result, which improves a previous result on the same topic, is proved. Theorem. Let \(q = p^h\) and \(1\leq p^e\leq q^{k-2}\), where \(e\) is a non-negative integer. If there are more than \(p^e(q-1)\) directions not determined by a set \(S\) of \(q^{k-1}\) points in \(AG(k, q)\) then every hyperplane meets \(S\) in \(0\) modulo \(p^{e+1}\) points. Constructions of point-sets which meet the bound in the case \(p^e = q^s\) and almost reach the bound when \(q= p^e\) are given in Section \(3\), and the paper ends with a section in which three conjectures arising from the above theorem (\textit{cylinder conjectures}) are stated.
reviewed by: V. Napolitano
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 51E99
zbMATH DE Number: 5536726
zbMATH Keywords: directions determined by a function
zbMATH Keywords: functions over finite fields
MaRDI profile type: Publication
full work available at URL: s10623-007-9108-z
OpenAlex ID: W2085970350
cites work: The number of directions determined by a function over a finite field.
cites work: On ovoids of parabolic quadrics
cites work: Q3594043
cites work: On the graph of a function in two variables over a finite field
cites work: On the number of slopes of the graph of a function defined on a finite field
cites work: Linear point sets and Rédei type \(k\)-blocking sets in \(\mathrm{PG}(n,q)\)
DOI: 10.1007/S10623-007-9108-Z
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description: scientific article; zbMATH DE number 5536726