688 research outputs found
Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights
We present a tree-based construction of LDPC codes that have minimum
pseudocodeword weight equal to or almost equal to the minimum distance, and
perform well with iterative decoding. The construction involves enumerating a
-regular tree for a fixed number of layers and employing a connection
algorithm based on permutations or mutually orthogonal Latin squares to close
the tree. Methods are presented for degrees and , for a
prime. One class corresponds to the well-known finite-geometry and finite
generalized quadrangle LDPC codes; the other codes presented are new. We also
present some bounds on pseudocodeword weight for -ary LDPC codes. Treating
these codes as -ary LDPC codes rather than binary LDPC codes improves their
rates, minimum distances, and pseudocodeword weights, thereby giving a new
importance to the finite geometry LDPC codes where .Comment: Submitted to Transactions on Information Theory. Submitted: Oct. 1,
2005; Revised: May 1, 2006, Nov. 25, 200
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
New Classes of Partial Geometries and Their Associated LDPC Codes
The use of partial geometries to construct parity-check matrices for LDPC
codes has resulted in the design of successful codes with a probability of
error close to the Shannon capacity at bit error rates down to . Such
considerations have motivated this further investigation. A new and simple
construction of a type of partial geometries with quasi-cyclic structure is
given and their properties are investigated. The trapping sets of the partial
geometry codes were considered previously using the geometric aspects of the
underlying structure to derive information on the size of allowable trapping
sets. This topic is further considered here. Finally, there is a natural
relationship between partial geometries and strongly regular graphs. The
eigenvalues of the adjacency matrices of such graphs are well known and it is
of interest to determine if any of the Tanner graphs derived from the partial
geometries are good expanders for certain parameter sets, since it can be
argued that codes with good geometric and expansion properties might perform
well under message-passing decoding.Comment: 34 pages with single column, 6 figure
Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder
This paper investigates the behavior of the Min-Sum decoder running on noisy
devices. The aim is to evaluate the robustness of the decoder in the presence
of computation noise, e.g. due to faulty logic in the processing units, which
represents a new source of errors that may occur during the decoding process.
To this end, we first introduce probabilistic models for the arithmetic and
logic units of the the finite-precision Min-Sum decoder, and then carry out the
density evolution analysis of the noisy Min-Sum decoder. We show that in some
particular cases, the noise introduced by the device can help the Min-Sum
decoder to escape from fixed points attractors, and may actually result in an
increased correction capacity with respect to the noiseless decoder. We also
reveal the existence of a specific threshold phenomenon, referred to as
functional threshold. The behavior of the noisy decoder is demonstrated in the
asymptotic limit of the code-length -- by using "noisy" density evolution
equations -- and it is also verified in the finite-length case by Monte-Carlo
simulation.Comment: 46 pages (draft version); extended version of the paper with same
title, submitted to IEEE Transactions on Communication
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