33 research outputs found
Degree Optimization and Stability Condition for the Min-Sum Decoder
The min-sum (MS) algorithm is arguably the second most fundamental algorithm
in the realm of message passing due to its optimality (for a tree code) with
respect to the {\em block error} probability \cite{Wiberg}. There also seems to
be a fundamental relationship of MS decoding with the linear programming
decoder \cite{Koetter}. Despite its importance, its fundamental properties have
not nearly been studied as well as those of the sum-product (also known as BP)
algorithm.
We address two questions related to the MS rule. First, we characterize the
stability condition under MS decoding. It turns out to be essentially the same
condition as under BP decoding. Second, we perform a degree distribution
optimization. Contrary to the case of BP decoding, under MS decoding the
thresholds of the best degree distributions for standard irregular LDPC
ensembles are significantly bounded away from the Shannon threshold. More
precisely, on the AWGN channel, for the best codes that we find, the gap to
capacity is 1dB for a rate 0.3 code and it is 0.4dB when the rate is 0.9 (the
gap decreases monotonically as we increase the rate).
We also used the optimization procedure to design codes for modified MS
algorithm where the output of the check node is scaled by a constant
. For , we observed that the gap to capacity was
lesser for the modified MS algorithm when compared with the MS algorithm.
However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates
between 0.3 and 0.9.
We conclude by posing what we consider to be the most important open
questions related to the MS algorithm.Comment: submitted to ITW 0
How to Find Good Finite-Length Codes: From Art Towards Science
We explain how to optimize finite-length LDPC codes for transmission over the
binary erasure channel. Our approach relies on an analytic approximation of the
erasure probability. This is in turn based on a finite-length scaling result to
model large scale erasures and a union bound involving minimal stopping sets to
take into account small error events. We show that the performances of
optimized ensembles as observed in simulations are well described by our
approximation. Although we only address the case of transmission over the
binary erasure channel, our method should be applicable to a more general
setting.Comment: 13 pages, 13 eps figures, enhanced version of an invited paperat the
4th International Symposium on Turbo Codes and Related Topics, Munich,
Germany, 200
Rate-Equivocation Optimal Spatially Coupled LDPC Codes for the BEC Wiretap Channel
We consider transmission over a wiretap channel where both the main channel
and the wiretapper's channel are Binary Erasure Channels (BEC). We use
convolutional LDPC ensembles based on the coset encoding scheme. More
precisely, we consider regular two edge type convolutional LDPC ensembles. We
show that such a construction achieves the whole rate-equivocation region of
the BEC wiretap channel.
Convolutional LDPC ensemble were introduced by Felstr\"om and Zigangirov and
are known to have excellent thresholds. Recently, Kudekar, Richardson, and
Urbanke proved that the phenomenon of "Spatial Coupling" converts MAP threshold
into BP threshold for transmission over the BEC.
The phenomenon of spatial coupling has been observed to hold for general
binary memoryless symmetric channels. Hence, we conjecture that our
construction is a universal rate-equivocation achieving construction when the
main channel and wiretapper's channel are binary memoryless symmetric channels,
and the wiretapper's channel is degraded with respect to the main channel.Comment: Working pape
Exchange of Limits: Why Iterative Decoding Works
We consider communication over binary-input memoryless output-symmetric
channels using low-density parity-check codes and message-passing decoding. The
asymptotic (in the length) performance of such a combination for a fixed number
of iterations is given by density evolution. Letting the number of iterations
tend to infinity we get the density evolution threshold, the largest channel
parameter so that the bit error probability tends to zero as a function of the
iterations.
In practice we often work with short codes and perform a large number of
iterations. It is therefore interesting to consider what happens if in the
standard analysis we exchange the order in which the blocklength and the number
of iterations diverge to infinity. In particular, we can ask whether both
limits give the same threshold.
Although empirical observations strongly suggest that the exchange of limits
is valid for all channel parameters, we limit our discussion to channel
parameters below the density evolution threshold. Specifically, we show that
under some suitable technical conditions the bit error probability vanishes
below the density evolution threshold regardless of how the limit is taken.Comment: 16 page
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard mode