This paper makes a deep study of regular two-distance sets. A set of unit vectors X in Euclidean space \({\mathbb {R}}^n\) is said to be regular two-distance set if the inner product of any pair of its vectors is either \(\alpha \) or \(\beta \), and the number of \(\alpha \)’s (and hence \(\beta \)’s) on each row of the Gram matrix of X is the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underlying space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. We also supply an example of a non-tight maximal two-distance frame. Connections among two-distance sets, equiangular lines, and quasi-symmetric designs are also discussed. For instance, we give a sufficient condition for constructing sets of equiangular lines from regular two-distance sets, especially from quasi-symmetric designs satisfying certain conditions.

中文翻译：

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常规两距离套装

本文对常规的两距离集进行了深入的研究。如果欧几里得空间\（{\ mathbb {R}} ^ n \）中的单位向量X的集合是任意对向量的内积是\（\ alpha \）或\（\测试\） ，和数量\（\阿尔法\）的（并且因此\（\测试\）的）上的革兰氏矩阵的每一行X是一样的 我们介绍了这些集合的各种属性，并着眼于它们为基础空间形成紧密框架的情况。然后，我们给出一些常规的两距离集的构造，尤其是两距离框架，包括紧紧情况和不紧紧情况。我们还提供了一个非紧密最大两距离帧的示例。还讨论了两距离集，等角线和准对称设计之间的连接。例如，我们给出了从规则的两距集构造等角线集的充分条件，尤其是从满足某些条件的拟对称设计中。