On Euclidean designs

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| ≤ .