Necessary Conditions for Existence of Some Designs in Polynomial Metric Spaces

Abstract

In this paper we consider designs in polynomial metric spaces with relatively small cardinalities (near to the classical bounds). We obtain restrictions on the distributions of the inner products of points of such designs. These conditions turn out to be strong enough to ensure obtaining nonexistence results already for the first open cases.

Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements

“…For a unified treatment of designs in terms of metric spaces consult the work of Levenshtein [48,49,50] (see also Ref. 's [51,52,53,54,55,56,57,58]). Our interest lies with the complex projective space CP d−1 of lines passing through the origin in C d .…”

Polynomial techniques for investigation of spherical designs

“…It is known (cf. [5]) that 4-designs of cardinality |C| < n(n + 1) does not posses a pair of antipodal points. Then for such designs we have calculation which is very similar to its analog from the previous subsection.…”

“…Then it follows from [12,Sect. 4] (see also [5]) that for every fixed cardinality |C| ≥ D(n, 2k − 1) there exist uniquely determined real numbers −1 ≤ α 0 < α 1 < · · · < α k−1 = s < 1 and ρ 0 , ρ 1 , . .…”

“…5. We investigate in detail the inner products of the special points (in particular these from Sect.…”

“…In [6,5,8] some partition of C into pairs was used for obtaining nonexistence when ρ 0 |C| < 2 (see Theorem 2.3). Now we see in more complicated situation that the attempt for partition of C into pairs and quadruples gives strong bounds on t |C|−2 (x) or t |C|−2 (z).…”