Maximal 2-distance sets containing the regular simplex
Maximal 2-distance sets containing the regular simplex
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包含正则单纯形的最大 2 距离集
DOI：
10.1016/j.disc.2020.112071
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发表时间：
2020
影响因子：
0.8
通讯作者：
Masashi Shinohara
中科院分区：
文献类型：
--
作者：
Hiroshi Nozaki;Masashi Shinohara
欧氏空间的有限子集X称为m-距离集，如果X中两个不同点之间的距离的数量等于m。一个m-距离集合X被称为极大的，如果在保持m-距离条件的情况下，任何向量都不能加到X上。本文研究了向量加到正则单形上的一个充要条件，使得该集合只有2个距离。我们构造了几个包含d维正则单形的d维极大2-距离集。特别地，存在无穷多个极大非球面2-距离集，它们包含正则单纯形和强可解设计的表示。最大2-距离集的大小为2 s 2（s+ 1），维数为d=（s− 1）（s+ 1）2− 1，其中s是素数幂。
A finite subset X of the Euclidean space is called an m-distance set if the number of distances between two distinct points in X is equal to m. An m-distance set X is said to be maximal if any vector cannot be added to X while maintaining the m-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only 2 distances. We construct several d-dimensional maximal 2-distance sets that contain a d-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical 2-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal 2-distance set has size 2 s 2 (s+ 1), and the dimension is d=(s− 1)(s+ 1) 2− 1, where s is a prime power.