A repository of spherical (t,t)-designs

Spherical (<em>t</em>,<em>t</em>)-designs in <strong>R</strong>^d are arrangements of points on the sphere S^d-1 (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.**R**^d中的球面(*t*,*t*)-设计（spherical (*t*,*t*)-designs）是单位球面S^d-1上的点集（可附带权重），其排布满足“彼此间距较远”：它们是有限点集，满足**R**^d中每个次数为2*t*的*d*元齐次多项式在球面上的积分，等于其在该点集上的平均值；该概念推广了球面*t*-设计与半设计的定义（相关内容可参见Conway与Sloane1993年著作的第3.3节），同时也推广了调和分析中的紧框架（tight frame）概念（Waldron, 2018）。针对复点集的情形，我们可将球面(*t*,*t*)-设计的定义推广至复球面(*t*,*t*)-设计：复(*d*-1)维球面上的有限点集（可附带权重），其满足对*d*元变量及其共轭分别齐次的多项式的积分条件，即变量总次数为*t*、共轭变量总次数亦为*t*。类似的定义也可推广至四元数与八元数域（Waldron, 2020）。如需获取更精确的讨论、历史背景以及既往结果与示例清单，请参阅参考文献列表。本数据集仓库包含各类球面(*t*,*t*)-设计与近设计——即极小化Waldron(2018)第6.16节所述设计势函数的点配置。本数据集的生成使用了Manopt软件（Boumal等人, 2014），其源代码可在aelzenaar/tightframes的GitHub仓库中获取。查看本数据集的最简方式为：下载index.html文件与四个.tgz压缩包，解压tar文件后将index.html与四个*_out目录置于同一目录下，随后在网页浏览器中打开index.html。设计数据可通过两种格式获取：Magma格式（文本文件）或.mat格式（可在Matlab中打开，设计数据存储于'result'变量中）。我们将在后续发表的论文中详细研究本数据集中出现的诸多新型设计。