Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes
type: Item
instance of: scholarly article
title: Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes
zbMATH Open document ID: 1178.51008
author: Simeon Ball
author: Szabolcs L. Fancsali
published in: Designs, Codes and Cryptography
publication date: 2009-11-04T00:00:00Z
review text: The Griesmer bound states that for any linear \([n, k, d]\) \(q\)-ary code \(C\), \[ n\geq g(k,d,q)=\sum_{i=0}^{k-1}\lceil d/q^i\rceil. \] In general, it is very interesting to investigate the properties of codes which attain this bound exactly, that is codes such that \(n = g(k, d, q)\). One of the main results of this paper is to show that for several values of \(d < q^{k-1}\) there are no such codes. In particular, the authors observe that any \([n, k]\)-code \(C\) with minimum distance \(d\leq q^{k-1}\) meeting the Griesmer bound is necessarily projective; that is no generator matrix for \(C\) has repeated columns. There is a close relationship between projective codes and \(t\)-fold blocking sets in \(\text{PG}(k-1, q)\); namely, if \(G\) is the \(k \times n\) generator matrix of a projective code C, then the set \(B\) of the \(|\text{PG}(k-1, q)|-n\) points of \(\text{PG}(k-1, q)\) which do not correspond to any of the columns of \(G\) forms a \(t\)-fold blocking set for the hyperplanes of \(\text{PG}(k-1, q)\), where \(t =|B|-q^{k-1} + d\). Suppose now \(B \subseteq \text{PG}(\delta, q)\) to be \(t\)-fold blocking set with respect to hyperplanes of \(\text{PG}(\delta, q)\) for which there actually exists a \(t\)-secant hyperplane. As a shorthand, write \[ t=[t_{\delta-2},t_{\delta-1},\dots,t_0]= \sum_{i=0}^{\delta-2}t_i\frac{q^{i+1}-1}{q-1} \] with \(0\leq t_i \leq q\) and \(t_0=t_1=\dots=t_j-1=0\) if \(t_j = q\). Then, the authors show that \[ |B|\leq qt+\sum_{i=0}^{\delta-2}t_i+q-1 \] implies \[ |B|\geq qt+\sum_{i=0}^{\delta-2}t_i. \] Furthermore, when \(q=p\) is a prime, they show that: {\parindent=5mm \begin{itemize}\item[1.] if \(B\) is a \([1, t_{\delta-3} , \ldots , t_0 ]\)-fold blocking set with respect to the hyperplanes of \(\text{PG}(\delta, p)\), \(\delta\geq 3\), \(t_i\leq p-1\) for all \(i\), \(t_{\delta-3}=0\) and \[ \sum_{i=0}^{\delta-3}t_i(p^{i+2}-1)<\frac{p-3}{2}(p^{\delta-1}-1)-(p-1)^2, \] then \[ |B|\geq tp+\sum_{i=0}^{\delta-2}t_i+p; \] \…
reviewed by: Luca Giuzzi
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 5625016
zbMATH Keywords: linear codes
zbMATH Keywords: Griesmer bound
zbMATH Keywords: \(t\)-fold blocking sets
zbMATH Keywords: minihypers
MaRDI profile type: Publication
full work available at URL: s10623-009-9298-7
OpenAlex ID: W2024761666
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DOI: 10.1007/S10623-009-9298-7
description: scientific article; zbMATH DE number 5625016