421 research outputs found
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
Short Codes with Mismatched Channel State Information: A Case Study
The rising interest in applications requiring the transmission of small
amounts of data has recently lead to the development of accurate performance
bounds and of powerful channel codes for the transmission of short-data packets
over the AWGN channel. Much less is known about the interaction between error
control coding and channel estimation at short blocks when transmitting over
channels with states (e.g., fading channels, phase-noise channels, etc...) for
the setup where no a priori channel state information (CSI) is available at the
transmitter and the receiver. In this paper, we use the mismatched-decoding
framework to characterize the fundamental tradeoff occurring in the
transmission of short data packet over an AWGN channel with unknown gain that
stays constant over the packet. Our analysis for this simplified setup aims at
showing the potential of mismatched decoding as a tool to design and analyze
transmission strategies for short blocks. We focus on a pragmatic approach
where the transmission frame contains a codeword as well as a preamble that is
used to estimate the channel (the codeword symbols are not used for channel
estimation). Achievability and converse bounds on the block error probability
achievable by this approach are provided and compared with simulation results
for schemes employing short low-density parity-check codes. Our bounds turn out
to predict accurately the optimal trade-off between the preamble length and the
redundancy introduced by the channel code.Comment: 5 pages, 5 figures, to appear in Proceedings of the IEEE
International Workshop on Signal Processing Advances in Wireless
Communications (SPAWC 2017
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
Tight bounds for LDPC and LDGM codes under MAP decoding
A new method for analyzing low density parity check (LDPC) codes and low
density generator matrix (LDGM) codes under bit maximum a posteriori
probability (MAP) decoding is introduced. The method is based on a rigorous
approach to spin glasses developed by Francesco Guerra. It allows to construct
lower bounds on the entropy of the transmitted message conditional to the
received one. Based on heuristic statistical mechanics calculations, we
conjecture such bounds to be tight. The result holds for standard irregular
ensembles when used over binary input output symmetric channels. The method is
first developed for Tanner graph ensembles with Poisson left degree
distribution. It is then generalized to `multi-Poisson' graphs, and, by a
completion procedure, to arbitrary degree distribution.Comment: 28 pages, 9 eps figures; Second version contains a generalization of
the previous resul
- …