6 research outputs found
A Generalization of the Finite-Length Scaling Approach Beyond the BEC
We want to extend the approximation of the error probability via a scaling approach from the BEC to general binary-input memoryless output-symmetric (BMS) channels. In particular, we consider such scaling laws for regular LDPC ensembles and message-passing (MP) decoders with a finite number of messages. We first show how to re-derive the scaling law for transmission over the BEC using an “EXIT-like” curve instead of the density evolution curve of the peeling decoder. The advantage of the new derivation is that the new expression of the scaling parameter α only contains quantities that can be meaningfully interpreted also for general message-passing algorithms. In particular, this expression only depends on the curvature of the EXIT-like curve as well as the variance of the messages, both taken at the critical channel parameter. We discuss how to compute these quantities for general MP algorithms and we evaluate the expressions for the specific cases of the Gallager algorithm A as well as the Decoder with Erasures and compare the resulting predictions on the error probability with simulation results
On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes
This paper addresses the issue of design of low-rate sparse-graph codes with
linear minimum distance in the blocklength. First, we define a necessary
condition which needs to be satisfied when the linear minimum distance is to be
ensured. The condition is formulated in terms of degree-1 and degree-2 variable
nodes and of low-weight codewords of the underlying code, and it generalizies
results known for turbo codes [8] and LDPC codes. Then, we present a new
ensemble of low-rate codes, which itself is a subclass of TLDPC codes [4], [5],
and which is designed under this necessary condition. The asymptotic analysis
of the ensemble shows that its iterative threshold is situated close to the
Shannon limit. In addition to the linear minimum distance property, it has a
simple structure and enjoys a low decoding complexity and a fast convergence.Comment: submitted to IEEE Trans. on Communication
Effects of Single-Cycle Structure on Iterative Decoding for Low-Density Parity-Check Codes
We consider communication over the binary erasure channel (BEC) using
low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For
fixed numbers of BP iterations, the bit error probability approaches a limit as
blocklength tends to infinity, and the limit is obtained via density evolution.
On the other hand, the difference between the bit error probability of codes
with blocklength and that in the large blocklength limit is asymptotically
where denotes a
specific constant determined by the code ensemble considered, the number of
iterations, and the erasure probability of the BEC. In this paper,
we derive a set of recursive formulas which allows evaluation of the constant
for standard irregular ensembles. The dominant difference
can be considered as effects of cycle-free and
single-cycle structures of local graphs. Furthermore, it is confirmed via
numerical simulations that estimation of the bit error probability using
is accurate even for small blocklengths.Comment: 16 pages, 7 figures, submitted to IEEE Transactions on Information
Theor
Tree-Structure Expectation Propagation for LDPC Decoding over the BEC
We present the tree-structure expectation propagation (Tree-EP) algorithm to
decode low-density parity-check (LDPC) codes over discrete memoryless channels
(DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be
used with any exponential family distribution over the cliques in the graph.
Second, it can impose additional constraints on the marginal distributions. We
use this second property to impose pair-wise marginal constraints over pairs of
variables connected to a check node of the LDPC code's Tanner graph. Thanks to
these additional constraints, the Tree-EP marginal estimates for each variable
in the graph are more accurate than those provided by BP. We also reformulate
the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type
algorithm (TEP) and we show that the algorithm has the same computational
complexity as BP and it decodes a higher fraction of errors. We describe the
TEP decoding process by a set of differential equations that represents the
expected residual graph evolution as a function of the code parameters. The
solution of these equations is used to predict the TEP decoder performance in
both the asymptotic regime and the finite-length regime over the BEC. While the
asymptotic threshold of the TEP decoder is the same as the BP decoder for
regular and optimized codes, we propose a scaling law (SL) for finite-length
LDPC codes, which accurately approximates the TEP improved performance and
facilitates its optimization