657 research outputs found
Modified augmented belief propagation for general memoryless channels
In this paper, we propose an efficient implementation of the augmented belief propagation (ABP) algorithm for low-density parity-check codes over general memoryless channels. ABP is a multistage BP based decoder that uses a backtracking processing when decoding fails. The algorithm proceeds in two main steps, namely a symbol selection step and an augmented decoding step. The former is based on a criterion related both to the stopping subgraph connectivity and to the input reliability, while the latter can be either implemented using a list based or a greedy approach. Compared to the original implementation, we consider a different approach for both steps. First, the proposed node selection is only based on the dynamic of sign changes of the extrinsic messages at the variable nodes output. This enables us to consider indifferently general memoryless channels, while still taking into account the graph irregularity. Then, we propose a simple yet efficient implementation of the augmented decoding procedure based on pruning of the branching tree The proposed algorithm shows near maximum likelihood decoding performance while decreasing the overall complexity (computation and memory) of the original algorithm. Moreover, complexity-performance trade-off is an built-in feature for this kind of algorithm
Modified belief propagation decoders applied to non-CSS QLDGM codes.
Quantum technology is becoming increasingly popular, and big companies are starting to invest huge amounts of money to ensure they do not get left behind in this technological race. Presently, qubits and operational quantum channels may be thought of as far-fetched ideas, but in the future, quantum computing will be of critical importance. In this project, it is provided a concise overview of the basics of coding theory and how they can be used in the design of quantum computers. Specifically, Low Density Parity Check (LDPC) codes are focused, as they can be integrated within the stabilizer construction to build effective quantum codes. Following this, it is introduced the specifics of the quantum paradigm and present the most common family of quantum codes: stabilizer codes. Finally, it is explained the codes that have been used in this project, discussing what type of code they are and how they are designed. In this last section, it is also presented the ultimate goal of the project: using modified belief propagation decoders that had previously been tested for QLDPCs, for the proposed non-CSS QLDGM codes of this project
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes
In this paper, we present an iterative soft-decision decoding algorithm for
Reed-Solomon codes offering both complexity and performance advantages over
previously known decoding algorithms. Our algorithm is a list decoding
algorithm which combines two powerful soft decision decoding techniques which
were previously regarded in the literature as competitive, namely, the
Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation
based on adaptive parity check matrices, recently proposed by Jiang and
Narayanan. Building on the Jiang-Narayanan algorithm, we present a
belief-propagation based algorithm with a significant reduction in
computational complexity. We introduce the concept of using a
belief-propagation based decoder to enhance the soft-input information prior to
decoding with an algebraic soft-decision decoder. Our algorithm can also be
viewed as an interpolation multiplicity assignment scheme for algebraic
soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200
Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance
Parameters of LDPC codes, such as minimum distance, stopping distance,
stopping redundancy, girth of the Tanner graph, and their influence on the
frame error rate performance of the BP, ML and near-ML decoding over a BEC and
an AWGN channel are studied. Both random and structured LDPC codes are
considered. In particular, the BP decoding is applied to the code parity-check
matrices with an increasing number of redundant rows, and the convergence of
the performance to that of the ML decoding is analyzed. A comparison of the
simulated BP, ML, and near-ML performance with the improved theoretical bounds
on the error probability based on the exact weight spectrum coefficients and
the exact stopping size spectrum coefficients is presented. It is observed that
decoding performance very close to the ML decoding performance can be achieved
with a relatively small number of redundant rows for some codes, for both the
BEC and the AWGN channels
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