Dense packings from quadratic fields and codes
type: Item
instance of: scholarly article
title: Dense packings from quadratic fields and codes
zbMATH Open document ID: 1162.52012
published in: Journal of Combinatorial Theory. Series A
publication date: 2008-09-04T00:00:00Z
review text: The author starts with the integral ring \(\mathcal{O}_{K}\) of the field \(K= \mathbb{Q}(\sqrt{3})\). This means \(\mathcal{O}_{K}=\{u+\omega v:u,v\in \mathbb{Z\}}\) with \(\omega =(-1+\sqrt{3})/2\), and so \(\mathcal{O}_{K}\) corresponds to the densest packing of congruent circles in \(\mathbb{R}^{2}\). Then he considers certain prime ideals in \(\mathcal{O}_{K}\) and shows how they can be concatenated with codes to produce dense packings in \(\mathbb{R} ^{n}\). For several values of \(n\) he gets packings with the best known densities. Let \(\Delta _{n}\) denote the maximal density of packings of congruent balls in \(\mathbb{R}^{n}\). Then \(\lambda :=\lim \sup_{n\rightarrow \infty }\frac{1}{n}\log _{2}(\Delta _{n})\geq -1\). This is the well-known Minkowski bound. It is shown that this might be improved to \(\lambda \geq -0.8471\), if the best upper bound in coding theory developed by \textit{Y. Ben-Haim} and \textit{S. Litsyn} [Adv. Math. Commun. 1, No. 1, 83--92 (2007; Zbl 1198.94199)] could be achieved.
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B99
zbMATH DE Number: 5321267
zbMATH Keywords: dense packings
zbMATH Keywords: asymptotic bounds
zbMATH Keywords: Minkowski bound
zbMATH Keywords: best-known packings
zbMATH Keywords: quadratic fields
author: Chaoping Xing
MaRDI profile type: Publication
full work available at URL: j.jcta.2007.12.005
OpenAlex ID: W2008106344
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DOI: 10.1016/J.JCTA.2007.12.005
description: scientific article; zbMATH DE number 5321267