544 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
Improved sphere packing lower bounds from Hurwitz lattices
In this paper we prove an asymptotic lower bound for the sphere packing
density in dimensions divisible by four. This asymptotic lower bound improves
on previous asymptotic bounds by a constant factor and improves not just lower
bounds for the sphere packing density, but also for the lattice sphere packing
density and, in fact, the Hurwitz lattice sphere packing density.Comment: 12 page
Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape
We extend the results from the first part of this series of two papers by
examining hyperuniformity in heterogeneous media composed of impenetrable
anisotropic inclusions. Specifically, we consider maximally random jammed
packings of hard ellipses and superdisks and show that these systems both
possess vanishing infinite-wavelength local-volume-fraction fluctuations and
quasi-long-range pair correlations. Our results suggest a strong generalization
of a conjecture by Torquato and Stillinger [Phys. Rev. E. 68, 041113 (2003)],
namely that all strictly jammed saturated packings of hard particles, including
those with size- and shape-distributions, are hyperuniform with signature
quasi-long-range correlations. We show that our arguments concerning the
constrained distribution of the void space in MRJ packings directly extend to
hard ellipse and superdisk packings, thereby providing a direct structural
explanation for the appearance of hyperuniformity and quasi-long-range
correlations in these systems. Additionally, we examine general heterogeneous
media with anisotropic inclusions and show for the first time that one can
decorate a periodic point pattern to obtain a hard-particle system that is not
hyperuniform with respect to local-volume-fraction fluctuations. This apparent
discrepancy can also be rationalized by appealing to the irregular distribution
of the void space arising from the anisotropic shapes of the particles. Our
work suggests the intriguing possibility that the MRJ states of hard particles
share certain universal features independent of the local properties of the
packings, including the packing fraction and average contact number per
particle.Comment: 29 pages, 9 figure
A bound on Grassmannian codes
We give a new asymptotic upper bound on the size of a code in the
Grassmannian space. The bound is better than the upper bounds known previously
in the entire range of distances except very large values.Comment: 5 pages, submitte
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