Universal Lower Bounds for Potential Energy of Spherical Codes

Abstract

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension n and cardinality N not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.

Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

“…This means that all designs on S n−1 of relatively small (fixed) cardinalities have their h-energy in very thin range. Indeed, our upper bounds are very close to the recently obtained universal lower bound [13,14]. As in [13,14,16] our results are valid for all absolutely monotone functions h.…”

“…which serves as a base for linear programming bounds for the cardinality and energy of spherical codes and designs (cf. [13,14,18,19,26,27]).…”

“…A spherical τ -design C ⊂ S n−1 as a nonempty finite set C ⊂ S n−1 Bounds for the energy of spherical designs on S 2 were obtained in [21,22] for particular h. The discrete Riesz s-energy of sequences of well separated τ -designs was investigated in [20]. General lower and upper bounds on energy of designs of fixed dimension, strength and cardinality were obtained by Boyvalenkov-Dragnev-Hardin-Saff-Stoyanova [13] (see also [14]).…”

Upper bounds for energies of spherical codes of given cardinality and separation

“…It is continuous and strictly increasing in s function, whose values at the endpoints of the intervals I m coincide with the Delsarte-Goethals-Seidel numbers D(n, m) (see [9] for the definition). Next, we introduce the notion of a 1/N -quadrature rule over subspaces consisting of polynomials (see [5]). The classical example of 1/N -quadrature rule is given by Levenshtein's Theorem 5.39 in [16], where a Gauss-Jacobi quadrature formula is defined (see [15] for the origin of this result).…”

“…It is clear from the above that M = L m (n, s) implies the coincidence of the upper and lower bounds in (26). In this case the corresponding codes are sharp configurations (also universally optimal codes; see [8]) which means that they attain simultaneously the Levenshtein bound, the ULB bound [5] and the upper bound from Theorem 3.2.…”