A family of low density matrices in Lagrangian-Grassmannian
type: Item
instance of: scholarly article
title: A family of low density matrices in Lagrangian-Grassmannian
zbMATH Open document ID: 1396.14043
author: Jesús Carrillo-Pacheco
author: F. Jarquín-Zárate
published in: Special Matrices
publication date: 2018-06-15T00:00:00Z
review text: This paper shows a relationship among the Lagrangian-Grassmannian variety \(\mathrm{LG}(n,2n)\) (which parametrizes the set of maximal isotropic subspaces of a symplectic vector space \(E\)\, of dimension \(2n\))\, defined over a finite field \(\mathbb{F}_q\)\, and the Low Density Parity-Check (LDPC) codes over \(\mathbb{F}_q\). In a previous work the first author of the present paper and \textit{F. Zaldivar} [Des. Codes Cryptography 60, No. 3, 291--298 (2011; Zbl 1225.94028)] proved that, over any field \(\mathbb{F}\), \(\mathrm{LG}(n,2n)=\mathrm{Gr}(n,2n)\cap Z(\Pi_{{\alpha}_{rs}})\),\, with \(\mathrm{Gr}(n,2n)\)\, the Grassmannian variety and \(\Pi_{{\alpha}_{rs}}\)\, a set of of linear homogeneous polynomials. That result is also improved now. Section 2 introduces, over any field \(\mathbb{F}\), the low density Boolean matrices \(\mathcal{L}_k\) and, based on them and according to a certain rule \(\mathcal{R}\), the low density Boolean matrix \(B\). Theorem 6 characterizes, in terms of \(\mathrm{char}(\mathbb{F})\), when those matrices have maximum rank. Section 3 shows that matrices \(\mathcal{L}_k\)\, define regular LDPC codes (Theorem 10) while matrix \(B\)\, define an irregular LDPC code (Theorem 12) (over a finite field \(\mathbb{F}_q\)). Section 4 proves that the family \(\Pi_{{\alpha}_{rs}}\)\, (see above) is unique up to linear combination (Theorem 16). Finally Proposition 16 gives an injective map \([E]\longrightarrow (\mathcal{R}, \mathcal{L}_r,\dots, \mathcal{L}_2)\), \(E\)\, a symplectic space, providing so a matrix invariant for \(E\).
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 15B34
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6890447
zbMATH Keywords: Lagrangian-Grassmannian varieties
zbMATH Keywords: Boolean matrices
zbMATH Keywords: low density parity-check code
reviewed by: Juan Tena Ayuso
MaRDI profile type: Publication
full work available at URL: spma-2018-0019
OpenAlex ID: W2805618366
cites work: On Lagrangian-Grassmannian codes
cites work: The Weight Distribution of a Family of Lagrangian-Grassmannian Codes
cites work: On codes over \(\mathrm{FFN}(1,q)\)-projective varieties
cites work: Q4302493
cites work: Q3293678
DOI: 10.1515/SPMA-2018-0019
description: scientific article; zbMATH DE number 6890447