On Euclidean tight 4-designs

Abstract

A spherical t -design is a finite subset X in the unit sphere S n - 1 ⊂ R n which replaces the value of the integral on the sphere of any polynomial of degree at most t by the average of the values of the polynomial on the finite subset X . Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean t -design in R n as a finite set X in R n for which ∑ i = 1 p ( w ( X i ) / ( | S i | ) ) ∫ S i f ( x ) d σ i ( x ) = ∑ x ∈ X w ( x ) f ( x ) holds for any polynomial f ( x ) of deg ( f ) ≤ t , where { S i , 1 ≤ i ≤ p } is the set of all the concentric spheres centered at the origin and intersect with X , X i = X ∩ S i , and w : X → R > 0 is a weight function of X . (The case of X ⊂ S n - 1 and with a constant weight corresponds to a spherical t -design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean 2 e -design. Let Y be a subset of R n and let 𝒫 e ( Y ) be the vector space consisting of all the polynomials restricted to Y whose degrees are at most e . Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that | X | ≥ dim ( 𝒫 e ( S ) ) holds, where S = ∪ i = 1 p S i . The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on S , the bound dim ( 𝒫 e ( S ) ) is natural and universal. In this point of view, we call a Euclidean 2 e -design X with | X | = dim ( 𝒫 e ( S ) ) a tight 2 e -design on p concentric spheres. Moreover if dim ( 𝒫 e ( S ) ) = dim ( 𝒫 e ( R n ) ) ( = n + e e ) holds, then we call X a Euclidean tight 2 e -design. We study the properties of tight Euclidean 2 e -designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in R n in the sense of Box and Hunter (1957) with the possible minimum size n + 2 2 . We also give examples of nontrivial Euclidean tight 4-designs in R 2 with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight 2 e -designs even for the nonconstant weight case for 2 e ≥ 4 .