Spherical two-distance sets

A subset X in the d-dimensional Euclidean space is called a kdistance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally kdistance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X, w) is a tight weighted spherical 2k-design. This result implies that locally kdistance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally twodistance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d − 1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d + 1)/2 points.We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.

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“…Musin proved this corollary by using a polynomial method in [20]. This corollary is used in proof of Theorem 4.13 in this paper.…”

Section: Corollary 34 Let X Be a Two-distance Set And A Innmentioning

confidence: 77%

“…Optimal twodistance sets in S 6 are given from three Chang graphs or the set of midpoints of edges of a regular simplex in R 7 . Moreover, Musin [20] determined D S * d (2) for 7 d < 40. …”

Section: Classifications Of Optimal Two-distance Setsmentioning

confidence: 99%

On a generalization of distance sets

Nozaki

Shinohara

2010

Journal of Combinatorial Theory, Series A

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“…This estimate was recently improved in [13] where it was shown that if the inner products between distinct code words take values t 1 , t 2 , and t 1 + t 2 ≥ 0, then |C| ≤ 1 /2n(n + 1). The proof relies on the method of linearly independent polynomials.…”

Section: Spherical Codesmentioning

confidence: 99%

“…Recently an improvement of the Delsarte-Goethals-Seidel bound on spherical s-codes for the case s = 2 was obtained in second author's paper [13]. Following this result, Nozaki [14] proved a general bound on the size of spherical s-codes.…”

Section: Introductionmentioning

confidence: 99%

Bounds on codes with few distances

Barg

Musin

2010

2010 IEEE International Symposium on Information Theory

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We prove a new bound on the size of codes with few distances in the Hamming space, improving an earlier result of P. Delsarte. We also improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets (constantweight codes) and the bound of Delsarte-Goethals-Seidel on the maximum size of spherical codes with few distances. Finally, we find the size of maximal binary codes and maximal constantweight codes of small length with 2,3, and 4 distances.

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“…Indeed we have a natural upper bound for the size of an s-distance set in R d , that is, |X| d+s s [2,4]. The largest cardinality of s-distance sets is known for several dimensions and s (see [7,9,12,20,23,25,26,31]). From the view point of distance sets, we want to obtain a small-dimensional representation of a graph.…”

Section: Introductionmentioning

confidence: 99%