More on the corner-vector construction for spherical designs

TL;DR

This paper generalizes the corner-vector method for constructing weighted spherical designs, establishing bounds, analyzing existence conditions, and providing examples linked to integral lattices.

Contribution

It introduces the generalized corner-vector method, extends theoretical bounds, and explores existence conditions with novel examples.

Findings

Established a uniform upper bound for design degrees.

Identified conditions for the existence of designs from the method.

Presented examples of designs characterized by integral lattices.

Abstract

This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the {\it generalized corner-vector method}. First we establish a uniform upper bound for the degree of designs obtained from the proposed method. Our proof is a hybrid argument that employs techniques in analysis and combinatorics, especially a famous result by Xu(1998) on the interrelation between spherical designs and simplical designs, and the cross-ratio comparison method for Hilbert identities introduced by Nozaki and Sawa(2013). We extensively study conditions for the existence of designs obtained from our method, and present many curious examples of degree $7$ through $13$, some of which are, to our surprise, characterized in terms of integral lattices.