5,333 research outputs found
Multiple Packing: Lower Bounds via Infinite Constellations
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . Given a well-known connection
with coding theory, multiple packings can be viewed as the Euclidean analog of
list-decodable codes, which are well-studied for finite fields. In this paper,
we derive the best known lower bounds on the optimal density of list-decodable
infinite constellations for constant under a stronger notion called
average-radius multiple packing. To this end, we apply tools from
high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
Multiple Packing: Lower and Upper Bounds
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . We study the multiple packing
problem for both bounded point sets whose points have norm at most
for some constant and unbounded point sets whose points are allowed to be
anywhere in . Given a well-known connection with coding theory,
multiple packings can be viewed as the Euclidean analog of list-decodable
codes, which are well-studied for finite fields. In this paper, we derive
various bounds on the largest possible density of a multiple packing in both
bounded and unbounded settings. A related notion called average-radius multiple
packing is also studied. Some of our lower bounds exactly pin down the
asymptotics of certain ensembles of average-radius list-decodable codes, e.g.,
(expurgated) Gaussian codes and (expurgated) spherical codes. In particular,
our lower bound obtained from spherical codes is the best known lower bound on
the optimal multiple packing density and is the first lower bound that
approaches the known large limit under the average-radius notion of
multiple packing. To derive these results, we apply tools from high-dimensional
geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
Multiple Packing: Lower Bounds via Error Exponents
We derive lower bounds on the maximal rates for multiple packings in
high-dimensional Euclidean spaces. Multiple packing is a natural generalization
of the sphere packing problem. For any and , a
multiple packing is a set of points in such that
any point in lies in the intersection of at most balls
of radius around points in . We study this problem
for both bounded point sets whose points have norm at most for some
constant and unbounded point sets whose points are allowed to be anywhere
in . Given a well-known connection with coding theory, multiple
packings can be viewed as the Euclidean analog of list-decodable codes, which
are well-studied for finite fields. We derive the best known lower bounds on
the optimal multiple packing density. This is accomplished by establishing a
curious inequality which relates the list-decoding error exponent for additive
white Gaussian noise channels, a quantity of average-case nature, to the
list-decoding radius, a quantity of worst-case nature. We also derive various
bounds on the list-decoding error exponent in both bounded and unbounded
settings which are of independent interest beyond multiple packing.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04407 and arXiv:2211.0440
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
Downlink Steered Space-Time Spreading Assisted Generalised Multicarrier DS-CDMA Using Sphere-Packing-Aided Multilevel Coding
This paper presents a novel generalised Multi-Carrier Direct Sequence Code Division Multiple Access (MC DS-CDMA) system invoking smart antennas for improving the achievable performance in the downlink, as well as employing multi-dimensional Sphere Packing (SP) modulation for increasing the achievable diversity product. In this contribution, the MC DS-CDMA transmitter considered employs multiple Antenna Arrays (AA) and each of the AAs consists of several antenna elements. Furthermore, the proposed system employs both time- and frequency- (TF) domain spreading for extending the achievable capacity, when combined with a novel user-grouping technique for reducing the effects of Multiuser Interference (MUI). Moreover, in order to further enhance the system’s performance, we invoke a MultiLevel Coding (MLC) scheme, whose component codes are determined using the so-called equivalent capacity based constituent-code rate-calculation procedure invoking a 4-dimensional bit-to-SP-symbol mapping scheme. Our results demonstrate an approximately 3.8 dB Eb/N0 gain over an identical throughput scheme dispensing with SP modulation at a BER of 10?5
Perfect, strongly eutactic lattices are periodic extreme
We introduce a parameter space for periodic point sets, given as unions of
translates of point lattices. In it we investigate the behavior of the
sphere packing density function and derive sufficient conditions for local
optimality. Using these criteria we prove that perfect, strongly eutactic
lattices cannot be locally improved to yield a periodic sphere packing with
greater density. This applies in particular to the densest known lattice sphere
packings in dimension and .Comment: 20 pages, 1 table; some corrections, incorporated referee suggestion
Turbo Detection of Symbol-Based Non-Binary LDPC-Coded Space-time Signals using Sphere Packing Modulation
A recently proposed space-time signal construction method that combines orthogonal design with sphere packing, referred to here as (STBC-SP), has shown useful performance improvements over Alamouti’s conventional orthogonal design. As a further advance, non-binary LDPC codes have been capable of attaining substantial performance improvements over their binary counterparts. In this paper, we demonstrate that the performance of STBC-SP systems can be further improved by concatenating sphere packing aided modulation with non-binary LDPC codes and performing symbolbased turbo detection. We present simulation results for the proposed scheme communicating over a correlated Rayleigh fading channel. At a BER of 10?6, the proposed symbolbased turbo-detected STBC-SP scheme was capable of achieving a coding gain of approximately 26.6dB over the identical throughput 1 bit/symbol uncoded STBC-SP benchmarker scheme. The proposed scheme also achieved a coding gain of approximately 3dB at a BER of 10?6 over a recently proposed bit-based turbo-detected STBC-SP benchmarker scheme
- …