Necessary conditions for existence of some designs in polynomial metric spaces
A nonempty finite code \(C \subset\) \(\mathcal M\), where \(\mathcal M\) is a polynomial metric space (PMS), is called a \(\tau\)-design if and only if the equality \(\sum_{x \in C}v(x) = 0\) holds for any function \(v(x) \in V_1 \oplus V_2 \oplus\cdots\oplus V_\tau\). This paper considers properties of such \(\tau\)-designs which possess a number of regularity properties. Known variously as spherical designs, classical \(t\)-designs, and orthogonal arrays, they are investigated in a variety of areas, including algebraic combinatorics, classical combinatorics and coding theory. The problem of finding lower bounds on the minimum possible size of designs in a PMS has been considered by \textit{P. Delsarte} [An algebraic approach to the association schemes in coding theory, Philips Res. Rep., Suppl. No. 10 (1973)] who obtained such a lower bound along with a number of other authors. A \(\tau\)-design \(\mathcal M\) in PMS is called tight if it attains this so-called Delsarte bound. Tight designs have been investigated by many authors, but seem to exist very rarely. Where nonexistence of tight designs is established the Delsarte bound, of course, is improved by one. In this paper the authors firstly obtain bounds on inner products of the points of \(\tau\)-designs, thereby giving necessary conditions for the existence of designs. These imply nonexistence results in many cases for both odd strengths and cardinalities. Bounds on the maximal inner product and the minimum distance of \(\tau\)-designs are obtained as well. The paper concludes with some applications of the results to the case of euclidean spheres.
scientific article; zbMATH DE number 1289995