Nonexistence of certain spherical designs of odd strengths and cardinalities
Let \(C\) be a finite nonempty subset of the Euclidean sphere \(S^{n-1}\). \(C\) is called a spherical \(T\)-design if, for all polynomials \(f\) of degree \(\leq T\), the average of \(f\) over the set is equal to the average of \(f\) over the sphere \(S^{n-1}\). The following lower bound on \(B(n,T)\), the minimum possible cardinality of a \(T\)-design on \(S^{n-1}\), is well known: \[ B(n,T)\geq R(n,T)=\begin{cases} {n+e-1\choose n-1} +{n+e-2\choose n-1} \quad & \text{if }T=2e;\\ 2{n+e-1 \choose n-1}\quad & \text{if }T= 2e+1.\end{cases} \] This paper deals primarily with the nonexistence of certain spherical designs with odd strengths and cardinalities. Necessary conditions for the existence of designs of odd strengths and cardinalities are presented which lead to nonexistence results in numerous cases. Designs with \(T=3\) and \(T=5\) are discussed in detail leading to the following results: Result I. There does not exist any spherical 3-design on \(S^{n-1}\) with \(R(n,3)+k=2n+k\) for all odd \(k<(2^{1\over 3}-1)n+p\) where \(p\approx 0.30018\). Results II. There exists no special 5-design on \(S^{n-1}\) with \(R(n,5)+k=n^2+n+k\) for all odd \(k<n^2 (2^{1\over 5}-1)/2+p_1n+p_2n\) where \(p_1\approx 0.0428\) and \(p_2\approx 0.0428\). Also the nonexistence of spherical 3-designs with seven points on \(S^2\) and 5-designs with 13 points on \(S^2\) is given. For situations where the nonexistence argument does not work, bounds on the minimum distance of such designs are presented.
scientific article; zbMATH DE number 1271014