249 research outputs found
Rateless Coding for Gaussian Channels
A rateless code-i.e., a rate-compatible family of codes-has the property that
codewords of the higher rate codes are prefixes of those of the lower rate
ones. A perfect family of such codes is one in which each of the codes in the
family is capacity-achieving. We show by construction that perfect rateless
codes with low-complexity decoding algorithms exist for additive white Gaussian
noise channels. Our construction involves the use of layered encoding and
successive decoding, together with repetition using time-varying layer weights.
As an illustration of our framework, we design a practical three-rate code
family. We further construct rich sets of near-perfect rateless codes within
our architecture that require either significantly fewer layers or lower
complexity than their perfect counterparts. Variations of the basic
construction are also developed, including one for time-varying channels in
which there is no a priori stochastic model.Comment: 18 page
Non-asymptotic Upper Bounds for Deletion Correcting Codes
Explicit non-asymptotic upper bounds on the sizes of multiple-deletion
correcting codes are presented. In particular, the largest single-deletion
correcting code for -ary alphabet and string length is shown to be of
size at most . An improved bound on the asymptotic
rate function is obtained as a corollary. Upper bounds are also derived on
sizes of codes for a constrained source that does not necessarily comprise of
all strings of a particular length, and this idea is demonstrated by
application to sets of run-length limited strings.
The problem of finding the largest deletion correcting code is modeled as a
matching problem on a hypergraph. This problem is formulated as an integer
linear program. The upper bound is obtained by the construction of a feasible
point for the dual of the linear programming relaxation of this integer linear
program.
The non-asymptotic bounds derived imply the known asymptotic bounds of
Levenshtein and Tenengolts and improve on known non-asymptotic bounds.
Numerical results support the conjecture that in the binary case, the
Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
Coherence Optimization and Best Complex Antipodal Spherical Codes
Vector sets with optimal coherence according to the Welch bound cannot exist
for all pairs of dimension and cardinality. If such an optimal vector set
exists, it is an equiangular tight frame and represents the solution to a
Grassmannian line packing problem. Best Complex Antipodal Spherical Codes
(BCASCs) are the best vector sets with respect to the coherence. By extending
methods used to find best spherical codes in the real-valued Euclidean space,
the proposed approach aims to find BCASCs, and thereby, a complex-valued vector
set with minimal coherence. There are many applications demanding vector sets
with low coherence. Examples are not limited to several techniques in wireless
communication or to the field of compressed sensing. Within this contribution,
existing analytical and numerical approaches for coherence optimization of
complex-valued vector spaces are summarized and compared to the proposed
approach. The numerically obtained coherence values improve previously reported
results. The drawback of increased computational effort is addressed and a
faster approximation is proposed which may be an alternative for time critical
cases
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