280 research outputs found
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Linear programming bounds for codes in Grassmannian spaces
We introduce a linear programming method to obtain bounds on the cardinality
of codes in Grassmannian spaces for the chordal distance. We obtain explicit
bounds, and an asymptotic bound that improves on the Hamming bound. Our
approach generalizes the approach originally developed by P. Delsarte and
Kabatianski-Levenshtein for compact two-point homogeneous spaces.Comment: 35 pages, 1 figur
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
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
On block coherence of frames
Block coherence of matrices plays an important role in analyzing the
performance of block compressed sensing recovery algorithms (Bajwa and Mixon,
2012). In this paper, we characterize two block coherence metrics: worst-case
and average block coherence. First, we present lower bounds on worst-case block
coherence, in both the general case and also when the matrix is constrained to
be a union of orthobases. We then present deterministic matrix constructions
based upon Kronecker products which obtain these lower bounds. We also
characterize the worst-case block coherence of random subspaces. Finally, we
present a flipping algorithm that can improve the average block coherence of a
matrix, while maintaining the worst-case block coherence of the original
matrix. We provide numerical examples which demonstrate that our proposed
deterministic matrix construction performs well in block compressed sensing
Coding for Errors and Erasures in Random Network Coding
The problem of error-control in random linear network coding is considered. A
``noncoherent'' or ``channel oblivious'' model is assumed where neither
transmitter nor receiver is assumed to have knowledge of the channel transfer
characteristic. Motivated by the property that linear network coding is
vector-space preserving, information transmission is modelled as the injection
into the network of a basis for a vector space and the collection by the
receiver of a basis for a vector space . A metric on the projective geometry
associated with the packet space is introduced, and it is shown that a minimum
distance decoder for this metric achieves correct decoding if the dimension of
the space is sufficiently large. If the dimension of each codeword
is restricted to a fixed integer, the code forms a subset of a finite-field
Grassmannian, or, equivalently, a subset of the vertices of the corresponding
Grassmann graph. Sphere-packing and sphere-covering bounds as well as a
generalization of the Singleton bound are provided for such codes. Finally, a
Reed-Solomon-like code construction, related to Gabidulin's construction of
maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum
distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification
Random Subsets of Structured Deterministic Frames have MANOVA Spectra
We draw a random subset of rows from a frame with rows (vectors) and
columns (dimensions), where and are proportional to . For a
variety of important deterministic equiangular tight frames (ETFs) and tight
non-ETF frames, we consider the distribution of singular values of the
-subset matrix. We observe that for large they can be precisely
described by a known probability distribution -- Wachter's MANOVA spectral
distribution, a phenomenon that was previously known only for two types of
random frames. In terms of convergence to this limit, the -subset matrix
from all these frames is shown to be empirically indistinguishable from the
classical MANOVA (Jacobi) random matrix ensemble. Thus empirically the MANOVA
ensemble offers a universal description of the spectra of randomly selected
-subframes, even those taken from deterministic frames. The same
universality phenomena is shown to hold for notable random frames as well. This
description enables exact calculations of properties of solutions for systems
of linear equations based on a random choice of frame vectors out of
possible vectors, and has a variety of implications for erasure coding,
compressed sensing, and sparse recovery. When the aspect ratio is small,
the MANOVA spectrum tends to the well known Marcenko-Pastur distribution of the
singular values of a Gaussian matrix, in agreement with previous work on highly
redundant frames. Our results are empirical, but they are exhaustive, precise
and fully reproducible
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