226 research outputs found
Equiangular Lines and Spherical Codes in Euclidean Space
A family of lines through the origin in Euclidean space is called equiangular
if any pair of lines defines the same angle. The problem of estimating the
maximum cardinality of such a family in was extensively studied
for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973,
in this paper we prove that for every fixed angle and sufficiently
large there are at most lines in with common angle
. Moreover, this is achievable only for . We
also show that for any set of fixed angles, one can find at most
lines in having these angles. This bound, conjectured by Bukh,
substantially improves the estimate of Delsarte, Goethals and Seidel from 1975.
Various extensions of these results to the more general setting of spherical
codes will be discussed as well.Comment: 24 pages, 0 figure
Grassmannian Frames with Applications to Coding and Communication
For a given class of uniform frames of fixed redundancy we define
a Grassmannian frame as one that minimizes the maximal correlation among all frames . We first analyze
finite-dimensional Grassmannian frames. Using links to packings in Grassmannian
spaces and antipodal spherical codes we derive bounds on the minimal achievable
correlation for Grassmannian frames. These bounds yield a simple condition
under which Grassmannian frames coincide with uniform tight frames. We exploit
connections to graph theory, equiangular line sets, and coding theory in order
to derive explicit constructions of Grassmannian frames. Our findings extend
recent results on uniform tight frames. We then introduce infinite-dimensional
Grassmannian frames and analyze their connection to uniform tight frames for
frames which are generated by group-like unitary systems. We derive an example
of a Grassmannian Gabor frame by using connections to sphere packing theory.
Finally we discuss the application of Grassmannian frames to wireless
communication and to multiple description coding.Comment: Submitted in June 2002 to Appl. Comp. Harm. Ana
New bounds for equiangular lines
A set of lines in is called equiangular if the angle between
each pair of lines is the same. We address the question of determining the
maximum size of equiangular line sets in , using semidefinite
programming to improve the upper bounds on this quantity. Improvements are
obtained in dimensions . In particular, we show that the
maximum number of equiangular lines in is for all and is 344 for This provides a partial resolution of the
conjecture set forth by Lemmens and Seidel (1973).Comment: Minor corrections; added one new reference. To appear in "Discrete
Geometry and Algebraic Combinatorics," A. Barg and O. R. Musin, Editors,
Providence: RI, AMS (2014). AMS Contemporary Mathematics serie
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