On Euclidean tight 4-designs

A spherical t-design is a finite subset X in the unit sphere Sn-1 subset of 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 Sigma(p)(i=1)(w(X-i)/(\S-i\)) integral(Si) f (x)d sigma(i)(x) = Sigma(x is an element of 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 boolean AND S-i, and w : X --> R->0 is a weight function of X. (The case of X subset of Sn-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 2e-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 = boolean OR S-p(i=1)i. The actual lower bounds proved by Delsarte and Seidel holds, where S = UP 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 2e-design X with \X\ = dim(P-e(S)) a tight 2e-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 2e-design. We study the properties of tight Euclidean 2e-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 2e-designs even for the nonconstant weight case for 2e >= 4.