Extremal polynomials for obtaining bounds for spherical codes and designs
type: Item
instance of: scholarly article
title: Extremal polynomials for obtaining bounds for spherical codes and designs
zbMATH Open document ID: 0836.94024
DOI: 10.1007/BF02570701
published in: Discrete \& Computational Geometry
publication date: 1996-05-08T00:00:00Z
full work available at URL: https://eudml.org/doc/131397
review text: A nonempty finite set \(W \subset S^{n - 1}\) is called a spherical code. Given \(|n |\geq 3\) and a maximal scalar product \(s\) (or a minimal distance \(d = \sqrt {2(1 - s)})\), it is interesting to know what is the maximal possible cardinality of a spherical code \(W \subset S^{n - 1}\), with \(s(W) \leq s\) (or \(d(W) \geq d)\). The maximal cardinality is denoted by \(T(n,s)\) i.e., \[ T(n,s) = \max \bigl \{|W |: W \subset S^{n - 1},\;(x,y) \leq s \text{ for all }x,y \in W,x \neq y \bigr\}. \] The number \(T(n,s)\) is equal to the maximum number of nonoverlapping spheres of radius \(r = d/(2 - d)\) (with \(d = \sqrt {2(1 - s)}\) as above) that can simultaneously touch the unit sphere \(S^{n - 1}\). A spherical code \(W \subset S^{n - 1}\) is called a spherical \(T\)- design if and only if \(\sum_{x \in W} f(x) = 0\) for all homogeneous harmonic polynomials \(f(x) = f(x_1, x_2, \dots, x_n)\) of degree \(1,2, \dots, T\). These designs were introduced by Delsarte et. al. in 1977. Seymour and Zaslavsky have shown the existence of spherical \(T\)- designs \(W \subset S^{n - 1}\) for any \(T\) and \(n\). Explicit constructions of spherical designs can be found. Given \(n \geq 3\) and an integer \(T \geq 4\) (the remaining cases are trivial) one wishes to find the minimal possible cardinality of a spherical \(T\) design on \(S^{n - 1}\). This problem has been solved by Delsarte et al. only in some cases. More precisely, for \(n \geq 3\), \(T \geq 4\) exactly eight designs are known to attain the Fisher type lower bound for their cardinality. Such designs are called tight. The tight designs were investigated e.g. by Bannai and Damerell. The upper bounds for spherical codes and the lower bounds for spherical designs were obtained by suitable polynomials. Now, there arise two problems: Given \(n \geq 3\), \(s \in (0,1)\) and integer \(k \geq 1\) one wishes to find a polynomial \(f(t) \in A_{n,s}\) of degree \(k\) (it it exists) which gives the best upper bound for \(T(n,s…
reviewed by: Ratnakaram Nava Mohan
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 784600
zbMATH Keywords: bounds
zbMATH Keywords: open problems
zbMATH Keywords: spherical code
zbMATH Keywords: extremal polynomials
zbMATH Keywords: spherical designs
author: Peter Boyvalenkov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784600