Extremal polynomials for obtaining bounds for spherical codes and designs
A nonempty finite set \(W \subset S^{n - 1}\) is called a spherical code. Given \(|n |\geq 3\) and a maximal scalar product \(s\) (or a minimal distance \(d = \sqrt {2(1 - s)})\), it is interesting to know what is the maximal possible cardinality of a spherical code \(W \subset S^{n - 1}\), with \(s(W) \leq s\) (or \(d(W) \geq d)\). The maximal cardinality is denoted by \(T(n,s)\) i.e., \[ T(n,s) = \max \bigl \{|W |: W \subset S^{n - 1},\;(x,y) \leq s \text{ for all }x,y \in W,x \neq y \bigr\}. \] The number \(T(n,s)\) is equal to the maximum number of nonoverlapping spheres of radius \(r = d/(2 - d)\) (with \(d = \sqrt {2(1 - s)}\) as above) that can simultaneously touch the unit sphere \(S^{n - 1}\). A spherical code \(W \subset S^{n - 1}\) is called a spherical \(T\)- design if and only if \(\sum_{x \in W} f(x) = 0\) for all homogeneous harmonic polynomials \(f(x) = f(x_1, x_2, \dots, x_n)\) of degree \(1,2, \dots, T\). These designs were introduced by Delsarte et. al. in 1977. Seymour and Zaslavsky have shown the existence of spherical \(T\)- designs \(W \subset S^{n - 1}\) for any \(T\) and \(n\). Explicit constructions of spherical designs can be found. Given \(n \geq 3\) and an integer \(T \geq 4\) (the remaining cases are trivial) one wishes to find the minimal possible cardinality of a spherical \(T\) design on \(S^{n - 1}\). This problem has been solved by Delsarte et al. only in some cases. More precisely, for \(n \geq 3\), \(T \geq 4\) exactly eight designs are known to attain the Fisher type lower bound for their cardinality. Such designs are called tight. The tight designs were investigated e.g. by Bannai and Damerell. The upper bounds for spherical codes and the lower bounds for spherical designs were obtained by suitable polynomials. Now, there arise two problems: Given \(n \geq 3\), \(s \in (0,1)\) and integer \(k \geq 1\) one wishes to find a polynomial \(f(t) \in A_{n,s}\) of degree \(k\) (it it exists) which gives the best upper bound for \(T(n,s)\) among the polynomials in \(A_{n,s}\) having the same or lower degree. Given \(n \geq 3\), \(T \geq 4\), and integer \(k \geq 1\) one wishes to find a polynomial \(f(t) \in B_{n,T}\) of degree \(k\) (if such polynomial exists) which gives the best lower bound for the cardinalities of the spherical \(T\) designs on \(S^{n - 1}\) among the polynomials in \(B_{n,T}\) having the same or lower degree. Definition: Any solution of the above two problems is called an \(A_{n,s}\)-extremal and \(B_{n,T}\)-extremal polynomial respectively. It is interesting to find extremal polynomials in order to obtain better bounds for spherical codes and designs. In the paper under review the two basic properties of extremal polynomials are investigated. First, a lower bound is given for the number of double zeros of extremal polynomials of high enough degree. Next, there is described a method of finding zero coefficients (if such exist) in the Gegenbauer expansion \[ f(t) = \sum^k_{i = 0} f_i P_i^{(n)} (t) \] of an extremal polynomial. This method, although sometimes carried out by a computer, works when the corresponding extremal polynomial exists and has zero Gegenbauer coefficients. The extremal polynomials (and new bounds respectively) or non-existence of extremal polynomials having the considered form are obtained. Thus the author has investigated two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs respectively. These results allow to search effectively using a computer. The best polynomials the author has obtained give substantial improvements in some cases on the previously known bounds for spherical codes and spherical designs, and give some examples with new upper bounds for spherical codes and new lower bounds for spherical designs. And lastly, the author has listed out some interesting open problems.
scientific article; zbMATH DE number 784600
scientific article