Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Symmetry

Joint Optimization of Four-Edge Type LDPC Codes with Symmetric Decoding Structure Based on EXIT Functions

Ying You, Guodong Su, Weiwei Lin

Published May 6, 2026• 0 citations
Publisher

Abstract

Four-edge type low-density parity-check (FET-LDPC) codes, as an important subclass of multi-edge type LDPC codes, offer greater design flexibility and performance potential due to their heterogeneous edge type structure. However, their multi-dimensional degree distribution significantly increases the complexity of optimization. This paper proposes a joint optimization framework for FET-LDPC codes leveraging the symmetric decoding structure inherent in their dual-branch architecture. The main contributions are as follows. First, an improved decoding model is established to analyze the mutual information transmission among the four edge types during iterative decoding, where the symmetry between the accumulator (ACC) and single parity-check (SPC) branches facilitates balanced information exchange. Second, a full-dimensional extrinsic information transfer (FEXIT) chart suitable for FET-LDPC codes is constructed, capturing the mutual information flow across branches. Third, a collaborative optimization model is designed by integrating the FEXIT chart with multi-constraint linear programming (LP) to perform asymmetric optimization for different edge types. The simulation results show that the proposed method achieves significant performance improvement over unoptimized codes in additive white Gaussian noise (AWGN) channels, particularly at a bit error rate (BER) of 10−6.

DOI
10.3390/sym18050794

Related papers

Efficient Design of MET-LDPC Code Ensembles via Multinomial Parameter Optimization

Linh Nguyen, Tuy Tan Nguyen

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.

IEEE Military Communications Conference (MILCOM)•Oct 6, 2025•0 citations

Complexity-optimized low-density parity-check codes for gallager decoding algorithm B

Wei Yu, Masoud Ardakani, B. Smith +1 more

The complexity-rate tradeoff for error-correcting codes below the Shannon limit is a central question in coding theory. This paper makes progress in this area by presenting a joint numerical optimization of rate and decoding complexity for low-density parity-check codes. The focus of this paper is on the binary symmetric channel and on a class of decoding algorithms for which an exact extrinsic information transfer (EXIT) chart analysis is possible. This class of decoding algorithms includes the Gallager decoding algorithm B. The main feature of the optimization method is a complexity measure based on the EXIT chart that accurately estimates the number of iterations required for the decoding algorithm to reach a target error rate. Under a fixed check-degree distribution, it is shown that the proposed complexity measure is a convex function of the variable-degree distribution in a region of interest. This allows us to numerically characterize the complexity-rate tradeoff. We show that for the Gallager B decoding algorithm on binary symmetric channels, the optimization procedure can produce complexity savings of 30-40% as compared to the conventional code design method

IEEE International Symposium on Information Theory (ISIT)•Jan 1, 2005•30 citations

On Design of Doubly-Generalized LDPC Codes Based on Multi-Type Information Functions

Enrico Paolini, Marco Chiani, M.P.C. Fossorier

Ensemble design of low-density parity-check (LDPC) codes and their generalizations is usually performed via numerical optimization techniques, such as differential evolution, in which a threshold analysis tool is always necessary. Threshold analysis of unstructured doubly-generalized LDPC (D-GLDPC) code ensembles over the binary erasure channel (BEC) can be performed via extrinsic information transfer (EXIT) chart, exploiting the information functions and split information functions of the check and variable component codes, respectively. In this paper, multi-type information functions of linear block codes are introduced as an extension of the concept of information functions, when the bit positions are assumed to be associated with different types. It is shown how multi-type information functions (together with their split counterparts) can be exploited within an EXIT analysis approach to perform threshold analysis over the BEC of multi-edge type D-GLDPC code ensembles. The proposed technique for threshold analysis captures D-GLDPC codes based on protographs as a special case.

Globecom•Dec 1, 2010•2 citations

On the design of good LDPC codes with joint genetic algorithm and linear programming optimization

Ahmadreza Amirzadeh, Mohamed Haj Taieb, Jean‐Yves Chouinard

In communication systems, the transmitted data is corrupted by channel perturbations, such as noise and fading, which affect the reliability of the received data. Error correction codes are employed to mitigate channel perturbations. However, design and implementation of good and efficient error correction codes remains an open problem. In this paper, Low Density Parity Check (LDPC) codes are considered as they provide a reasonable trade-off between computational complexity and reliability. Good LDPC codes should ideally provide low complexity, close to capacity acheivable transmission rate, high coding threshold, and high decoding stability. In this paper, we investigate a joint LDPC code optimization algorithm using Genetic Algorithm (GA) and Linear Programming (LP) to determine the variable nodes and check nodes degrees distributions. EXIT chart analysis and Frame Error Rate (FER) performance are used to validate the proposed method.

Canadian Workshop on Information Theory (CWIT)•Jun 1, 2017•2 citations

On the EXIT chart analysis of low-density parity-check codes

S.R. Kollu, Hamid Jafarkhani

The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.

Globecom•Jan 1, 2005•14 citations

Parameter-Coupled Offset Min-Sum Decoding with Edge-Type Differentiation for MET-LDPC Codes

Ying You, Guodong Su, Weiwei Lin

To improve the decoding performance of multi-edge type low-density parity-check (MET-LDPC) codes, this paper proposes an edge-type differentiated parameter coupling offset min-sum (EDPC-OMS) decoding algorithm. The contributions are threefold. First, we replace the traditional uniform compensation with edge-type differentiated compensation, resolving the mismatch between the decoding model and code structure. Second, we introduce a parameter coupling mechanism that enables joint optimization of multiple edge types while maintaining differentiated configurations. Third, a practically feasible design combining precomputation and look-up tables enables dynamic parameter adjustment with moderate additional overhead, achieving a favorable performance–complexity trade-off. Simulation results over additive white Gaussian noise (AWGN) channels and Rayleigh fading channels demonstrate that the proposed algorithm adaptively selects offset factors according to channel conditions and edge types without introducing significant computational complexity, effectively lowering the bit error rate and enhancing decoding capability.

Mathematics•Apr 17, 2026•1 citation
Four-edge type low-density parity-check (FET-LDPC) codes, as an important subclass of multi-edge type LDPC codes, offer greater design flexibility and performance potential due to their heterogeneous edge type structure. However, their multi-dimensional degree distribution significantly increases the complexity of optimization. This paper proposes a joint optimization framework for FET-LDPC codes leveraging the symmetric decoding structure inherent in their dual-branch architecture. The main contributions are as follows. First, an improved decoding model is established to analyze the mutual information transmission among the four edge types during iterative decoding, where the symmetry between the accumulator (ACC) and single parity-check (SPC) branches facilitates balanced information exchange. Second, a full-dimensional extrinsic information transfer (FEXIT) chart suitable for FET-LDPC codes is constructed, capturing the mutual information flow across branches. Third, a collaborative optimization model is designed by integrating the FEXIT chart with multi-constraint linear programming (LP) to perform asymmetric optimization for different edge types. The simulation results show that the proposed method achieves significant performance improvement over unoptimized codes in additive white Gaussian noise (AWGN) channels, particularly at a bit error rate (BER) of 10−6.
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.
The complexity-rate tradeoff for error-correcting codes below the Shannon limit is a central question in coding theory. This paper makes progress in this area by presenting a joint numerical optimization of rate and decoding complexity for low-density parity-check codes. The focus of this paper is on the binary symmetric channel and on a class of decoding algorithms for which an exact extrinsic information transfer (EXIT) chart analysis is possible. This class of decoding algorithms includes the Gallager decoding algorithm B. The main feature of the optimization method is a complexity measure based on the EXIT chart that accurately estimates the number of iterations required for the decoding algorithm to reach a target error rate. Under a fixed check-degree distribution, it is shown that the proposed complexity measure is a convex function of the variable-degree distribution in a region of interest. This allows us to numerically characterize the complexity-rate tradeoff. We show that for the Gallager B decoding algorithm on binary symmetric channels, the optimization procedure can produce complexity savings of 30-40% as compared to the conventional code design method
Ensemble design of low-density parity-check (LDPC) codes and their generalizations is usually performed via numerical optimization techniques, such as differential evolution, in which a threshold analysis tool is always necessary. Threshold analysis of unstructured doubly-generalized LDPC (D-GLDPC) code ensembles over the binary erasure channel (BEC) can be performed via extrinsic information transfer (EXIT) chart, exploiting the information functions and split information functions of the check and variable component codes, respectively. In this paper, multi-type information functions of linear block codes are introduced as an extension of the concept of information functions, when the bit positions are assumed to be associated with different types. It is shown how multi-type information functions (together with their split counterparts) can be exploited within an EXIT analysis approach to perform threshold analysis over the BEC of multi-edge type D-GLDPC code ensembles. The proposed technique for threshold analysis captures D-GLDPC codes based on protographs as a special case.
In communication systems, the transmitted data is corrupted by channel perturbations, such as noise and fading, which affect the reliability of the received data. Error correction codes are employed to mitigate channel perturbations. However, design and implementation of good and efficient error correction codes remains an open problem. In this paper, Low Density Parity Check (LDPC) codes are considered as they provide a reasonable trade-off between computational complexity and reliability. Good LDPC codes should ideally provide low complexity, close to capacity acheivable transmission rate, high coding threshold, and high decoding stability. In this paper, we investigate a joint LDPC code optimization algorithm using Genetic Algorithm (GA) and Linear Programming (LP) to determine the variable nodes and check nodes degrees distributions. EXIT chart analysis and Frame Error Rate (FER) performance are used to validate the proposed method.
The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.
To improve the decoding performance of multi-edge type low-density parity-check (MET-LDPC) codes, this paper proposes an edge-type differentiated parameter coupling offset min-sum (EDPC-OMS) decoding algorithm. The contributions are threefold. First, we replace the traditional uniform compensation with edge-type differentiated compensation, resolving the mismatch between the decoding model and code structure. Second, we introduce a parameter coupling mechanism that enables joint optimization of multiple edge types while maintaining differentiated configurations. Third, a practically feasible design combining precomputation and look-up tables enables dynamic parameter adjustment with moderate additional overhead, achieving a favorable performance–complexity trade-off. Simulation results over additive white Gaussian noise (AWGN) channels and Rayleigh fading channels demonstrate that the proposed algorithm adaptively selects offset factors according to channel conditions and edge types without introducing significant computational complexity, effectively lowering the bit error rate and enhancing decoding capability.