An Upper Bound for the Cardinality of an s -Distance Set in Euclidean Space

Bannaĭ, Eiichi; Kawasaki, Kazuki; Nitamizu, Yusuke; Sato, Teppei

doi:10.1007/s00493-003-0032-1

Cited by 26 publications

(6 citation statements)

References 7 publications

(6 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The upper bound n(n + 3)/2 for spherical two-distance sets [5], the bound n+2 2 for Euclidean two-distance sets [2], as well as the bound n+s s for s−distance sets [1,3] were obtained by the polynomial method. The main idea of this method is the following: to associate sets to polynomials and show that these polynomials are linearly independent as members of the corresponding vector space.…”

Section: Linearly Independent Polynomialsmentioning

confidence: 99%

Spherical two-distance sets

Musin

2009

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g(n) of spherical two-distance sets does not exceed n(n + 3)/2. This upper bound is known to be tight for n = 2, 6, 22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n) = n(n + 1)/2 for g(n).In this paper using the so-called polynomial method it is proved that for nonnegative a + b the largest cardinality of S is not greater than L(n). For the case a + b < 0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n) = L(n) for 6 < n < 22, 23 < n < 40, and g(23) = 276 or 277.

show abstract

Section: Linearly Independent Polynomialsmentioning

confidence: 99%

Spherical two-distance sets

Musin

2009

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…Bannai, Bannai and Stanton [2] and Blokhuis [3] showed independently that the cardinality of a s-distance set in E n does not exceed n+s s , so we have the same upper bound for the cardinalities of both isosceles sets and 2-distance sets.…”

Section: Introductionmentioning

confidence: 60%

Isosceles Sets

Ionin¹

2009

Electron. J. Combin.

View full text Add to dashboard Cite

In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\choose 2}$ for the Euclidean space ${\Bbb E}^{n}$ was found by Blokhuis. This upper bound is known to be sharp for $n=1$, 2, 6, and 8. We will consider Erdős' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\leq{n+1\choose 2}+1$; if, furthermore, $n\geq 4$, $n\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\leq{n\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\leq{n\choose 2}+1$. The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for ${\Bbb E}^{n}$ with $n\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.

show abstract

“…Next, assume that V = ∪ p i=1 S i , where the S i are spheres in R n . E. Bannai, K. Kawasaki, Y. Nitamizu, and T. Sato proved the following result in [5,Theorem 1] for the case when the spheres S i are concentric. We have a much shorter approach to the same bound, in a more general setting, without the assumption on the centers.…”

Section: A| ≤mentioning

confidence: 99%