2,795 research outputs found
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View
These are the notes for a set of lectures delivered by the two authors at the
Les Houches Summer School on `Complex Systems' in July 2006. They provide an
introduction to the basic concepts in modern (probabilistic) coding theory,
highlighting connections with statistical mechanics. We also stress common
concepts with other disciplines dealing with similar problems that can be
generically referred to as `large graphical models'.
While most of the lectures are devoted to the classical channel coding
problem over simple memoryless channels, we present a discussion of more
complex channel models. We conclude with an overview of the main open
challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July
2006, 44 pages, 25 ps figure
Numerical Techniques for Finding the Distances of Quantum Codes
We survey the existing techniques for calculating code distances of classical
codes and apply these techniques to generic quantum codes. For classical and
quantum LDPC codes, we also present a new linked-cluster technique. It reduces
complexity exponent of all existing deterministic techniques designed for codes
with small relative distances (which include all known families of quantum LDPC
codes), and also surpasses the probabilistic technique for sufficiently high
code rates.Comment: 5 pages, 1 figure, to appear in Proceedings of ISIT 2014 - IEEE
International Symposium on Information Theory, Honolul
Entanglement-assisted quantum turbo codes
An unexpected breakdown in the existing theory of quantum serial turbo coding
is that a quantum convolutional encoder cannot simultaneously be recursive and
non-catastrophic. These properties are essential for quantum turbo code
families to have a minimum distance growing with blocklength and for their
iterative decoding algorithm to converge, respectively. Here, we show that the
entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in
the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can
possess both of the aforementioned desirable properties. We give several
examples of EAQ convolutional encoders that are both recursive and
non-catastrophic and detail their relevant parameters. We then modify the
quantum turbo decoding algorithm of Poulin et al., in order to have the
constituent decoders pass along only "extrinsic information" to each other
rather than a posteriori probabilities as in the decoder of Poulin et al., and
this leads to a significant improvement in the performance of unassisted
quantum turbo codes. Other simulation results indicate that
entanglement-assisted turbo codes can operate reliably in a noise regime 4.73
dB beyond that of standard quantum turbo codes, when used on a memoryless
depolarizing channel. Furthermore, several of our quantum turbo codes are
within 1 dB or less of their hashing limits, so that the performance of quantum
turbo codes is now on par with that of classical turbo codes. Finally, we prove
that entanglement is the resource that enables a convolutional encoder to be
both non-catastrophic and recursive because an encoder acting on only
information qubits, classical bits, gauge qubits, and ancilla qubits cannot
simultaneously satisfy them.Comment: 31 pages, software for simulating EA turbo codes is available at
http://code.google.com/p/ea-turbo/ and a presentation is available at
http://markwilde.com/publications/10-10-EA-Turbo.ppt ; v2, revisions based on
feedback from journal; v3, modification of the quantum turbo decoding
algorithm that leads to improved performance over results in v2 and the
results of Poulin et al. in arXiv:0712.288
Self-concatenated code design and its application in power-efficient cooperative communications
In this tutorial, we have focused on the design of binary self-concatenated coding schemes with the help of EXtrinsic Information Transfer (EXIT) charts and Union bound analysis. The design methodology of future iteratively decoded self-concatenated aided cooperative communication schemes is presented. In doing so, we will identify the most important milestones in the area of channel coding, concatenated coding schemes and cooperative communication systems till date and suggest future research directions
Sparse Graph Codes for Quantum Error-Correction
We present sparse graph codes appropriate for use in quantum
error-correction. Quantum error-correcting codes based on sparse graphs are of
interest for three reasons. First, the best codes currently known for classical
channels are based on sparse graphs. Second, sparse graph codes keep the number
of quantum interactions associated with the quantum error correction process
small: a constant number per quantum bit, independent of the blocklength.
Third, sparse graph codes often offer great flexibility with respect to
blocklength and rate. We believe some of the codes we present are unsurpassed
by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened
version was resubmitted to IEEE Transactions on Information Theory Jan 20,
200
Binary Message Passing Decoding of Product-like Codes
We propose a novel binary message passing decoding algorithm for product-like
codes based on bounded distance decoding (BDD) of the component codes. The
algorithm, dubbed iterative BDD with scaled reliability (iBDD-SR), exploits the
channel reliabilities and is therefore soft in nature. However, the messages
exchanged by the component decoders are binary (hard) messages, which
significantly reduces the decoder data flow. The exchanged binary messages are
obtained by combining the channel reliability with the BDD decoder output
reliabilities, properly conveyed by a scaling factor applied to the BDD
decisions. We perform a density evolution analysis for generalized low-density
parity-check (GLDPC) code ensembles and spatially coupled GLDPC code ensembles,
from which the scaling factors of the iBDD-SR for product and staircase codes,
respectively, can be obtained. For the white additive Gaussian noise channel,
we show performance gains up to dB and dB for product and
staircase codes compared to conventional iterative BDD (iBDD) with the same
decoder data flow. Furthermore, we show that iBDD-SR approaches the performance
of ideal iBDD that prevents miscorrections.Comment: Accepted for publication in the IEEE Transactions on Communication
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
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