LDPC codes associated with linear representations of geometries
type: Item
instance of: scholarly article
title: LDPC codes associated with linear representations of geometries
zbMATH Open document ID: 1218.51007
published in: Advances in Mathematics of Communications
publication date: 2011-06-03T00:00:00Z
review text: Low density parity check matrices, introduced by \textit{R. G. Gallager} [IRE Trans. Inform. Theory IT-8, 21--28 (1962; Zbl 0107.11802)], are codes whose parity check matrices have a small percentage of nonzero elements. These codes have received a great deal of attention recently because they often outperform other codes and sometimes are very near the Shannon bound. The author examines LDPC codes over a field \(K\) constructed from finite geometries \(T^*_2({\mathcal K})\), where \({\mathcal K}\) is any subset of \(PG(2,q)\), \(q=p^h\), \(p \neq \text{char}(K).\) To define \(T^*_2({\mathcal K})\) let \(\Pi\) be a hyperplane in 3-dimensional projective space over \(F_q\). In \(T^*_2({\mathcal K})\), the points are the points of \(PG(3,q) \setminus PG(2,q)\), the lines are the affine lines which are incident with a point of \({\mathcal K}\) and the incidence relation is that of the ambient space. Examples include \(LU(3,q)^D\), the generalized quadrangle \(T^*_2({\mathcal K})\) with \({\mathcal K}\) a hyperoval, the affine space \(AG(3,q)\), and several partial and semi-partial geometries. The author gives a general setting for this study and provides a unified approach to previous results. In particular, he proves an expression for the dimension and minimum weight of the code, classifies code words of minimum weight, and proves that the code is generated by words of minimum weight.
reviewed by: Steven T. Dougherty
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 5903539
zbMATH Keywords: LDPC codes
zbMATH Keywords: linear codes
zbMATH Keywords: finite geometry
zbMATH Keywords: linear representation
zbMATH Keywords: minimum distance
zbMATH Keywords: dimension
author: P. Vandendriessche
describes a project that uses: LDPC
MaRDI profile type: Publication
full work available at URL: amc.2010.4.405
OpenAlex ID: W2026448191
DOI: 10.3934/AMC.2010.4.405
Recommended article: Some low-density parity-check codes derived from finite geometries
Recommended article: Low-density parity-check codes based on finite geometries: a rediscovery and new results
Recommended article: Linear representations of finite geometries and associated LDPC codes
Recommended article: Q4460826
Recommended article: On Algebraic Construction of Gallager and Circulant Low-Density Parity-Check Codes
Recommended article: Q3558596
Recommended article: Q3464682
Recommended article: LDPC codes generated by conics in the classical projective plane
Recommended article: Small weight codewords in the LDPC codes arising from linear representations of geometries
description: scientific article; zbMATH DE number 5903539