Characterization of \(\{2(q+1) + 2, 2; t, q\}\)-minihypers in \(PG(t,q) (t\geqslant 3, q\in \{3,4\})\)
type: Item
instance of: scholarly article
title: Characterization of \(\{2(q+1) + 2, 2; t, q\}\)-minihypers in \(PG(t,q) (t\geqslant 3, q\in \{3,4\})\)
zbMATH Open document ID: 0776.51011
DOI: 10.1016/0012-365X(93)90487-E
author: Noboru Hamada
author: Tor Helleseth
author: Øyvind Ytrehus
published in: Discrete Mathematics
publication date: 1993-06-20T00:00:00Z
review text: An \(\{f,m;t,q\}\)-minihyper is a set \(F\) of \(f\) points in \(PG(t,q)\) such that \(F\) meets every hyperplane in at least \(m\) points and at least one hyperplane in exactly \(m\) points. Minihypers have connections with codes and the problem of characterizing all \([n,k,d,q]\)-codes (such a code is a \(k\)-dimensional subspace of an \(n\)-dimensional vector space over \(GF(q)\) such that every nonzero vector has Hamming weight at least \(d)\) meeting the Griesmer bound, i.e. such that \(n=\sum^{k-1}_{i=0}\left[{d\over q^ i}\right]\), lead the authors to the problem of characterizing the minihypers in the title. Under other restrictions, a characterization exists already (e.g. \(t\geq 3\) and \(q\geq 5\) was done by the first author and \textit{M. Deza} [Discrete Math. 71, No. 3, 219-231 (1988; Zbl 0678.94010)]). The main result of the paper under review is the fact that there exist no irreducible \(\{12,2;t,4\}\)-minihypers and that there exists exactly one class of irreducible \(\{10,2;t,3\}\)-minihypers. Irreducible means that the minihyper is not a union of at least two disjoint minihypers.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 205599
zbMATH Keywords: codes
zbMATH Keywords: minihypers
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
cites work: A Bound for Error-Correcting Codes
cites work: Characterization of \(\{(q+1)+2,1;t,q\}-\min \cdot hypers\) and \(\{2(q+1)+2,2;2,q\}-\min \cdot hypers\) in a finite projective geometry
cites work: Characterization of \(\{2(q+1)+2,2;t,q\}\)-\(\min \cdot hypers\) in PG(t,q) (t\(\geq 3,q\geq 5)\) and its applications to error-correcting codes
full work available at URL: 0012-365x(93)90487-e
OpenAlex ID: W2019432609
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description: scientific article; zbMATH DE number 205599