446 research outputs found
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
A method for dense packing discovery
The problem of packing a system of particles as densely as possible is
foundational in the field of discrete geometry and is a powerful model in the
material and biological sciences. As packing problems retreat from the reach of
solution by analytic constructions, the importance of an efficient numerical
method for conducting \textit{de novo} (from-scratch) searches for dense
packings becomes crucial. In this paper, we use the \textit{divide and concur}
framework to develop a general search method for the solution of periodic
constraint problems, and we apply it to the discovery of dense periodic
packings. An important feature of the method is the integration of the unit
cell parameters with the other packing variables in the definition of the
configuration space. The method we present led to improvements in the
densest-known tetrahedron packing which are reported in [arXiv:0910.5226].
Here, we use the method to reproduce the densest known lattice sphere packings
and the best known lattice kissing arrangements in up to 14 and 11 dimensions
respectively (the first such numerical evidence for their optimality in some of
these dimensions). For non-spherical particles, we report a new dense packing
of regular four-dimensional simplices with density
and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure
Asymptotic bounds for spherical codes
The set of all error-correcting codes C over a fixed finite alphabet F of
cardinality q determines the set of code points in the unit square with
coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal
distance). The central problem of the theory of such codes consists in
maximizing simultaneously the transmission rate of the code and the relative
minimum Hamming distance between two different code words. The classical
approach to this problem explored in vast literature consists in the inventing
explicit constructions of "good codes" and comparing new classes of codes with
earlier ones. Less classical approach studies the geometry of the whole set of
code points (R,delta) (with q fixed), at first independently of its
computability properties, and only afterwords turning to the problems of
computability, analogies with statistical physics etc. The main purpose of this
article consists in extending this latter strategy to domain of spherical
codes.Comment: 34 pages amstex, 3 figure
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Sphere packings revisited
AbstractIn this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows:–Hadwiger numbers of convex bodies and kissing numbers of spheres;–touching numbers of convex bodies;–Newton numbers of convex bodies;–one-sided Hadwiger and kissing numbers;–contact graphs of finite packings and the combinatorial Kepler problem;–isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture;–the strong Kepler conjecture;–bounds on the density of sphere packings in higher dimensions;–solidity and uniform stability.Each topic is discussed in details along with some of the “most wanted” research problems
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Random Sequential Addition of Hard Spheres in High Euclidean Dimensions
Employing numerical and theoretical methods, we investigate the structural
characteristics of random sequential addition (RSA) of congruent spheres in
-dimensional Euclidean space in the infinite-time or
saturation limit for the first six space dimensions ().
Specifically, we determine the saturation density, pair correlation function,
cumulative coordination number and the structure factor in each =of these
dimensions. We find that for , the saturation density
scales with dimension as , where and
. We also show analytically that the same density scaling
persists in the high-dimensional limit, albeit with different coefficients. A
byproduct of this high-dimensional analysis is a relatively sharp lower bound
on the saturation density for any given by , where is the structure factor at
(i.e., infinite-wavelength number variance) in the high-dimensional limit.
Consistent with the recent "decorrelation principle," we find that pair
correlations markedly diminish as the space dimension increases up to six. Our
work has implications for the possible existence of disordered classical ground
states for some continuous potentials in sufficiently high dimensions.Comment: 38 pages, 9 figures, 4 table
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