On Euclidean tight 4-designs

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Abstract

A spherical t-design is a finite subset X in the unit sphere Sn-1⊂Rn 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 Rn as a finite set X in Rn for which $¥sum$i=1p(w(Xi)/(|Si|)) ∫S<sub>i</sub>f(x)dσi(x) = $¥sum$x∈Xw(x)f(x) holds for any polynomial f(x) of deg(f)≤t, where {Si, 1≤i≤p} is the set of all the concentric spheres centered at the origin and intersect with X, Xi=X∩Si, and w:X→R>0 is a weight function of X. (The case of X⊂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 Rn and let $¥mathscr{P}$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($¥mathscr{P}$e(S)) holds, where S=∪i=1pSi. 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($¥mathscr{P}$e(S)) is natural and universal. In this point of view, we call a Euclidean 2e-design X with |X| = dim($¥mathscr{P}$e(S)) a tight 2e-design on p concentric spheres. Moreover if dim($¥mathscr{P}$e(S)) = dim($¥mathscr{P}$e(Rn)) (=${n+e ¥choose e}$) 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 Rn in the sense of Box and Hunter (1957) with the possible minimum size ${n+2 ¥choose 2}$. We also give examples of nontrivial Euclidean tight 4-designs in R2 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.

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