A generalization of Larman-Rogers-Seidel's theorem

A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set is greater than 2d+3, then there exists an integer k such that a^2/b^2=(k-1)/k, where a and b are the distances. In this paper, we give an extension of this theorem for any s. Namely, if the size of an s-distance set is greater than some value depending on d and s, then certain functions of s distances become integers. Moreover, we prove that if the size of X is greater than the value, then the number of s-distance sets is finite.Comment: 12 pages, no figur

A remark on sets with few distances in ℝ^d

A celebrated theorem due to Bannai-Bannai-Stanton says that if A is a set of points in ℝ^d, which determines s distinct distances, then |A| ≤ (d+s/s). In this note, we give a new simple proof of this result by combining Sylvester's Law of Inertia for quadratic forms with the proof of the so-called Croot-Lev-Pach Lemma from additive combinatorics

Bounds on three- and higher-distance sets

A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.Comment: 12 page

Complex spherical codes with two inner products

A finite set $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly regular tournaments, or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric $D$-optimal designs. To prove them, we show the smallest embedding dimension of a tournament into a complex sphere by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel matrix.Comment: 10 pages, to appear in European Journal of Combinatoric