On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre
type: Item
instance of: scholarly article
title: On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre
zbMATH Open document ID: 0654.94014
DOI: 10.1007/BF01393742
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143577
review text: Let C be a code of length n over an alphabet A of size q, \(q\geq 2\), \(| A| =q\). If no two words agree in k positions and \(| C| =q\) k then C is called a maximum distance separable (MDS) code. A question of interest is, given k,q what is the maximum value of n for which such codes exist? In the case when C is linear over \(F=GF(q)\) it can be generated by a \(k\times n\) matrix X over F of rank k, in which every set of k columns are independent. The columns of X are regarded as points in a projective space of dimension k-1, PG(k-1,q). The columns of X then form a k-arc in the space. Thus questions of s-arcs in the space relate directly to MDS codes. The following questions were posed by \textit{B. Segre} [Ann. Mat. pura appl., IV. Ser. 39, 357-379 (1955; Zbl 0066.140)]: a) For given r and q what is the maximum value of s for which there exist s-arcs in PG(r,q). b) For what values of r and q, with \(q>r+1\), is every \((q+1)\)-arc of PG(r,q) the point set of a normal rational curve? c) For given r, and q, with \(q>r+1\), what are the values of s for which every s-arc of PG(r,q) is contained in a normal rational curve of this space? Substantial progress is reported here on these three problems with attention confined to the case of q even.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 4068780
zbMATH Keywords: maximum distance separable code
zbMATH Keywords: projective space
zbMATH Keywords: MDS codes
zbMATH Keywords: arcs
zbMATH Keywords: normal rational curve
author: Aiden A. Bruen
author: Joseph A. Thas
author: Aart Blokhuis
reviewed by: Ian Blake
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4068780