Affiliation:
1. Department of Mathematics, Gettysburg College, 300 North Washington Street, Gettysburg, PA 17325, USA.
Abstract
Abstract
A Euclidean t-design, as introduced by Neumaier and Seidel (1988), is a finite set 𝒳 ⊂ ℝ
n
with a weight function w : 𝒳 → ℝ+ for which
holds for every polynomial ƒ of total degree at most t; here R is the set of norms of the points in 𝒳, Wr
is the total weight of all elements of 𝒳 with norm r, Sr
is the n-dimensional sphere of radius r centered at the origin, and
is the average of ƒ over Sr
. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), also proved a Fisher-type inequality |𝒳| ≥ N(n, |R|, t) (assuming that the design is antipodal if t is odd). For fixed n and |R| we have N(n, |R|, t) = O(t
n−1).
This paper contains two main results. First, we provide a recursive construction for Euclidean t-designs in ℝ
n
. Namely, we show how to use certain Gauss–Jacobi quadrature formulae to ‘lift’ a Euclidean t-design in ℝ
n−1 to a Euclidean t-design in ℝ
n
, preserving both the norm spectrum R and the weight sum Wr
for each r ∈ R. For fixed n and |R| this construction yields a design of size O(t
n−1); however, the coefficient of t
n−1 here is significantly greater than it is in N(n, |R|, t).
A Euclidean design with exactly N(n, |R|, t) points is called tight; in both of the above mentioned papers it was conjectured that a tight Euclidean design with t ≥ 4 must be a spherical design, that is, |R| = 1 and w is constant on 𝒳. Bannai and Bannai (2003) disproved this conjecture by exhibiting an example for the parameters, (n, |R|, t) = (2,2,4). Here we construct tight Euclidean designs for n = 2 and every t and |R| with |R| ≤
.
Cited by
12 articles.
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