167 research outputs found
Improved Delsarte bounds for spherical codes in small dimensions
We present an extension of the Delsarte linear programming method. For
several dimensions it yields improved upper bounds for kissing numbers and for
spherical codes. Musin's recent work on kissing numbers in dimensions three and
four can be viewed in our framework.Comment: 16 pages, 3 figures. Substantial changes after referee's comments,
one new lemm
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
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
Sphere packing bounds via spherical codes
The sphere packing problem asks for the greatest density of a packing of
congruent balls in Euclidean space. The current best upper bound in all
sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We
revisit their argument and improve their bound by a constant factor using a
simple geometric argument, and we extend the argument to packings in hyperbolic
space, for which it gives an exponential improvement over the previously known
bounds. Additionally, we show that the Cohn-Elkies linear programming bound is
always at least as strong as the Kabatiansky-Levenshtein bound; this result is
analogous to Rodemich's theorem in coding theory. Finally, we develop
hyperbolic linear programming bounds and prove the analogue of Rodemich's
theorem there as well.Comment: 30 pages, 2 figure
New upper bounds for kissing numbers from semidefinite programming
Recently A. Schrijver derived new upper bounds for binary codes using
semidefinite programming. In this paper we adapt this approach to codes on the
unit sphere and we compute new upper bounds for the kissing number in several
dimensions. In particular our computations give the (known) values for the
cases n = 3, 4, 8, 24.Comment: 17 pages, (v4) references updated, accepted in Journal of the
American Mathematical Societ
On kissing numbers and spherical codes in high dimensions
We prove a lower bound of on the
kissing number in dimension . This improves the classical lower bound of
Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a
similar linear factor improvement to the best known lower bound on the maximal
size of a spherical code of acute angle in high dimensions
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