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IEEE International Symposium on Information Theory (ISIT)

Design of multi-edge type bilayer-expurgated LDPC codes

Marwan Hadri Azmi, Jinhong Yuan

Published Jun 1, 2009• 12 citations
Publisher

Abstract

This paper presents a new bilayer-expurgated LDPC (BE-LDPC) code design for decode-and-forward relay system. The code design is based on the multi-edge type LDPC (MET-LDPC) code structure. The lower and upper graphs' edges of the BE-LDPC codes are represented via different edge types. We call the codes as multi-edge type bilayer-expurgated LDPC (MET-BE-LDPC) codes. We derive the relationship between the lower and the overall graphs of the MET-BE-LDPC codes using the multi-edge type multinomials. This relationship acts as a constraint in optimizing the MET-BE-LDPC codes under the density evolution. We apply the differential evolution algorithm to search for the best MET-BE-LDPC codes. Applying the proposed methodology, two MET-BE-LDPC codes are designed. The codes perform asymptotically within 0.08474-0.5087 dB from the theoretical limits. Finally, we demonstrate that the proposed methodology can be used to design the BE-LDPC codes at a larger source-to-relay and source-to-destination SNR differences.

DOI
10.1109/isit.2009.5205732

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This paper presents a new bilayer-expurgated LDPC (BE-LDPC) code design for decode-and-forward relay system. The code design is based on the multi-edge type LDPC (MET-LDPC) code structure. The lower and upper graphs' edges of the BE-LDPC codes are represented via different edge types. We call the codes as multi-edge type bilayer-expurgated LDPC (MET-BE-LDPC) codes. We derive the relationship between the lower and the overall graphs of the MET-BE-LDPC codes using the multi-edge type multinomials. This relationship acts as a constraint in optimizing the MET-BE-LDPC codes under the density evolution. We apply the differential evolution algorithm to search for the best MET-BE-LDPC codes. Applying the proposed methodology, two MET-BE-LDPC codes are designed. The codes perform asymptotically within 0.08474-0.5087 dB from the theoretical limits. Finally, we demonstrate that the proposed methodology can be used to design the BE-LDPC codes at a larger source-to-relay and source-to-destination SNR differences.
A novel optimization algorithm to design a bilayer lengthened QC-LDPC code for relay channel is proposed in this paper, which combine differential evolution algorithm and bilayer graph for LDPC codes. We aim to construct QC-LDPC codes with best girth(maximum girth and minimum numbers of shortest cycles)both at relay node and destination node. Proposed algorithm is composed of two steps, first the exponent matrix of QC-LDPC code for lower graph is searched by differential evolution algorithm, then the exponent matrix of QC-LDPC code for overall graph is also found by differential evolution algorithm based on the obtained exponent matrix for lower graph. Simulation results verify the validity of proposed algorithm. Constructed QC-LDPC codes may perform well both at relay node and destination node.
Designing multi-edge type low-density parity-check (MET-LDPC) code ensembles that satisfy structural constraints while achieving near-capacity performance remains a challenging problem. Existing methods, including those based on differential evolution, have improved flexibility but still face limitations such as high computational cost and the need for manual configuration of edge patterns. We propose an optimization-based method that explores diverse, structurally feasible edge distribution candidates and optimizes node distributions via a constrained linear system formulation. A dual annealing optimizer is then applied to identify node profiles that maximize the decoding threshold. Evaluations on both the binary erasure channel and the binary-input additive white Gaussian noise channel across multiple code rates show that our method consistently produces ensembles with strong threshold performance and low edge complexity, highlighting its practicality and adaptability.
This paper considers the optimization of multi-edge type low-density parity-check (METLDPC) codes to maximize the decoding threshold. We propose an algorithm to jointly optimize the node degree distribution and the multi-edge structure of MET-LDPC codes for given values of the maximum number of edge-types and maximum node degrees. This joint optimization is particularly important for MET-LDPC codes as it is not clear a priori which structures will be good. Using several examples, we demonstrate that the MET-LDPC codes designed by the proposed joint optimization algorithm exhibit improved decoding thresholds compared to previously reported MET-LDPC codes.
This letter proposes a fast optimization procedure to design bilayer-expurgated low density parity check codes in the relay channel. The code optimization problem aims to maximize the code rate while guaranteeing the convergence of the density evolution relations in upper and lower layer codes. The original problem is linear programming. Nevertheless, it suffers from high computational complexity, as it is semi-infinite programming (SIP) in a 2-D continuous interval. To efficiently solve it, we first approximate the density evolutions by polynomials. Then, leveraging optimization problems over the solution of sum of square multivariate polynomials the problem is transformed to semi-definite programming (SDP), where the global solution can be found efficiently using available SDP solvers. Our simulation shows that our approach results in significantly lower computational complexity with respect to the well-known method of quantization for solving the SIP problem, while both achieve the same performance.
This paper considers density evolution for low-density parity-check (LDPC) and multi-edge type LDPC (MET-LDPC) codes over the binary input additive white Gaussian noise channel. We first analyze three single-parameter Gaussian approximations for density evolution and discuss their accuracy under several conditions, namely, at low rates, with punctured and degree-one variable nodes. We observe that the assumption of symmetric Gaussian distribution for the density-evolution messages is not accurate in the early decoding iterations, particularly at low rates and with punctured variable nodes. Thus, single-parameter Gaussian approximation methods produce very poor results in these cases. Based on these observations, we then introduce a new density evolution approximation algorithm for LDPC and MET-LDPC codes. Our method is a combination of full density evolution and a single-parameter Gaussian approximation, where we assume a symmetric Gaussian distribution only after density-evolution messages closely follow a symmetric Gaussian distribution. Our method significantly improves the accuracy of the code threshold estimation. Additionally, the proposed method significantly reduces the computational time of evaluating the code threshold compared with full density evolution thereby making it more suitable for code design.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.