New asymptotic bounds on the size of multiple packings of the Euclidean sphere
type: Item
instance of: scholarly article
title: New asymptotic bounds on the size of multiple packings of the Euclidean sphere
zbMATH Open document ID: 1235.52028
author: Vladimir Blinovsky
author: Simon Litsyn
published in: Discrete \& Computational Geometry
publication date: 2011-11-25T00:00:00Z
review text: A finite set (spherical code) \(K^n \subset \mathbb{S}^{n-1}\) is an \(L\)-multiple packing of radius \(t\) if \(\max_{x \in \mathbb{R}^n} |B(x,t) \cup K^n| \leq L\). Asymptotic estimations on the maximal possible size of an \(L\)-multiple packing are stated as relations between the code rate \(R\), defined by \(\ln |K^n| \sim Rn\) and \[ t_L= \lim\sup_{n \to \infty} \sup_{|K^n| \geq exp(Rn)} \frac{\sup t_L(K^n)}{n}, \] where \(t_L\) is the supremum of \(t\) for the code \(K^n\). In the paper under review, the authors obtain new upper bounds on \(t_L\). The proof is based on linear programming bounds on the distance distributions of large spherical codes followed by a recursive procedure for finding a small simplex of \(L+1\) points of \(K^n\).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 94B75
Mathematics Subject Classification ID: 90C05
zbMATH DE Number: 5979922
zbMATH Keywords: sphere packings
zbMATH Keywords: multiple packings
zbMATH Keywords: spherical codes
reviewed by: Peter Boyvalenkov
MaRDI profile type: Publication
full work available at URL: s00454-011-9336-x
OpenAlex ID: W2124291447
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DOI: 10.1007/S00454-011-9336-X
description: scientific article; zbMATH DE number 5979922