Necessary conditions for existence of some designs in polynomial metric spaces
type: Item
instance of: scholarly article
title: Necessary conditions for existence of some designs in polynomial metric spaces
zbMATH Open document ID: 0929.05008
author: Silvia Boumova
author: Peter Boyvalenkov
author: Danyo Danev
published in: European Journal of Combinatorics
publication date: 2000-01-11T00:00:00Z
review text: A nonempty finite code \(C \subset\) \(\mathcal M\), where \(\mathcal M\) is a polynomial metric space (PMS), is called a \(\tau\)-design if and only if the equality \(\sum_{x \in C}v(x) = 0\) holds for any function \(v(x) \in V_1 \oplus V_2 \oplus\cdots\oplus V_\tau\). This paper considers properties of such \(\tau\)-designs which possess a number of regularity properties. Known variously as spherical designs, classical \(t\)-designs, and orthogonal arrays, they are investigated in a variety of areas, including algebraic combinatorics, classical combinatorics and coding theory. The problem of finding lower bounds on the minimum possible size of designs in a PMS has been considered by \textit{P. Delsarte} [An algebraic approach to the association schemes in coding theory, Philips Res. Rep., Suppl. No. 10 (1973)] who obtained such a lower bound along with a number of other authors. A \(\tau\)-design \(\mathcal M\) in PMS is called tight if it attains this so-called Delsarte bound. Tight designs have been investigated by many authors, but seem to exist very rarely. Where nonexistence of tight designs is established the Delsarte bound, of course, is improved by one. In this paper the authors firstly obtain bounds on inner products of the points of \(\tau\)-designs, thereby giving necessary conditions for the existence of designs. These imply nonexistence results in many cases for both odd strengths and cardinalities. Bounds on the maximal inner product and the minimum distance of \(\tau\)-designs are obtained as well. The paper concludes with some applications of the results to the case of euclidean spheres.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 1289995
zbMATH Keywords: polynomial metric space
zbMATH Keywords: spherical design
zbMATH Keywords: \(\tau\)-design
zbMATH Keywords: self-complementary design
zbMATH Keywords: antipodal design
zbMATH Keywords: Delsarte bound
reviewed by: Peter B. Gibbons
MaRDI profile type: Publication
full work available at URL: eujc.1998.0278
OpenAlex ID: W1969059239
DOI: 10.1006/EUJC.1998.0278
description: scientific article; zbMATH DE number 1289995