DOI: 10.5281/zenodo.6443357
title: A repository of spherical (t,t)-designs

description: Spherical (<em>t</em>,<em>t</em>)-designs in <strong>R</strong><sup>d</sup> are arrangements of points on the sphere S<sup>d-1</sup> (possibly with weights) which are spaced "far apart from each other": they are finite sets in such that the integral over the sphere of each homogeneous polynomial of degree 2<em>t</em> in <em>d</em> variables is equal to its average value on the set, and generalise the notion of a spherical <em>t</em>-design and half-design which can be found for example in Section 3.3 of Conway and Sloane (1993), and the notion of a tight frame from harmonic analysis (Waldron, 2018). There is a generalisation of the definition of a spherical (<em>t</em>,<em>t</em>)-design to complex point arrangements: a complex spherical (<em>t</em>,<em>t</em>)-design is a finite set on the complex (<em>d</em>-1)-sphere (again, possibly with weights) which integrates polynomials which are separately homogeneous in <em>d</em> variables and their conjugates, such that the total degree in the variables is <em>t</em> and the total degree in the conjugate variables is also <em>t</em>. Similar definitions can also be made over the quaternions and octonions (Waldron, 2020). For a more precise discussion, history, and a list of prior results and examples see the references list. This repository is a set of files containing various spherical (<em>t</em>,<em>t</em>)-designs and near-designs - point configurations which minimise the design potential function of Section 6.16 of Waldron (2018). These files were produced using the Manopt software (Boumal et. al., 2014), and the source code may be found in the aelzenaar/tightframes GitHub repository. The easiest way to view the dataset is to download index.html and the four .tgz files; decompress the tar files so that index.html is in the same directory as the four *_out directories, and open index.html in a web browser. The design itself can then be found in either Magma format (a text file) or .mat format (open in Matlab, and then the design is found in the 'result' variable). In a forthcoming paper we will study in detail many of the new designs which appear here.
description: {"references": ["N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre. Manopt, a Matlab toolbox for optimization on manifolds. In: Journal of Machine Learning Research 15.42 (2014), pp. 1455-1459. URL: https://www.manopt.org/", "Jennifer Bramwell. \"On the existence of spherical (t,t)-designs\". BSc(Hons) dissertation. The University of Auckland, 2011. URL: https://www.math.auckland.ac.nz/~waldron/Students/Jennifer/JenniferBramwelldissertation.pdf", "John H. Conway and Neil J.A. Sloane. Sphere packings, lattices, and groups. 2nd ed. Grundlehren der mathematischen Wissenschaften 290. Springer-Verlag, 1988.", "Phillippe Delsarte, J.-M. Goethals, and Johan Jacob Seidel. \"Spherical codes and designs\". In: Geometriae Dedicatae 6.3 (1977), pp.363-388.", "Alex Elzenaar, \"Numerical construction of spherical (t,t)-designs\". Summer Research Scholarship project report. The University of Auckland, 2020. URL: https://aelzenaar.github.io/tight_frames_project/scholarship_report.pdf", "Daniel Hughes and Shayne Waldron. \"Spherical (t,t)-designs with a small number of vectors\". In: Involve 13.2 (2020), pp.193-203.", "Mozhgan Mohammadpour and Shayne Waldron. Constructing high order spherical designs as a union of two of lower order. 2019. arXiv:1912.07151 [math.MG]", "Shayne Waldron. An introduction to finite tight frames. Applied and numerical harmonic analysis. Birkh\u00e4user, 2018.", "Shayne Waldron. Tight frames over the quaternions and equiangular lines. 2020. Preprint: https://www.math.auckland.ac.nz/~waldron/Preprints/Quaternions/quaternions.html"]}