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arXiv:cs/0502084

On the Typicality of the Linear Code Among the LDPC Coset Code Ensemble

Chih-Chun Wang, H. Vincent Poor, Sanjeev R. Kulkarni

Published Feb 22, 2005• 4 citations
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Abstract

Density evolution (DE) is one of the most powerful analytical tools for low-density parity-check (LDPC) codes on memoryless binary-input/symmetric-output channels. The case of non-symmetric channels is tackled either by the LDPC coset code ensemble (a channel symmetrizing argument) or by the generalized DE for linear codes on non-symmetric channels. Existing simulations show that the bit error rate performances of these two different approaches are nearly identical. This paper explains this phenomenon by proving that as the minimum check node degree $d_c$ becomes sufficiently large, the performance discrepancy of the linear and the coset LDPC codes is theoretically indistinguishable. This typicality of linear codes among the LDPC coset code ensemble provides insight into the concentration theorem of LDPC coset codes.

DOI
10.48550/arxiv.cs/0502084

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Density evolution (DE) is one of the most powerful analytical tools for low-density parity-check (LDPC) codes on memoryless binary-input/symmetric-output channels. The case of non-symmetric channels is tackled either by the LDPC coset code ensemble (a channel symmetrizing argument) or by the generalized DE for linear codes on non-symmetric channels. Existing simulations show that the bit error rate performances of these two different approaches are nearly identical. This paper explains this phenomenon by proving that as the minimum check node degree dcd_cdc​ becomes sufficiently large, the performance discrepancy of the linear and the coset LDPC codes is theoretically indistinguishable. This typicality of linear codes among the LDPC coset code ensemble provides insight into the concentration theorem of LDPC coset codes.
Density evolution (DE) is one of the most powerful analytical tools for low-density parity-check (LDPC) codes on memoryless binary-input/symmetric-output channels. The case of non-symmetric chan-nels is tackled either by the LDPC coset code ensemble (a channel symmetrizing argument) or by the gen-eralized DE for linear codes on non-symmetric channels. Existing simulations show that the performance discrepancy of these two different approaches is nearly identical. This paper explains this phenomenon by proving that as the minimum check node degree dc becomes sufficiently large, the performance discrep-ancy of the linear and the coset LDPC codes is theoretically indistinguishable. This typicality of linear codes among the LDPC coset code ensemble provides insight into the concentration theorem of LDPC coset codes.
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 has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian 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, binary asymmetric 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
Abstract — 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. Index Terms — Low-density parity-check (LDPC) codes, density evolution, sum-product algorithm, asymmetric channels, z-channels, rank of random matrices. I.
Density evolution (DE) 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 has been widely and successfully applied to different channels, including binary erasure channels (BECs), binary symmetric channels (BSCs), binary additive white Gaussian noise (BiAWGN) channels, etc. This paper generalizes density evolution for asymmetric 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 (BP) 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.
This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.
Decoding performance of LDPC (Low-Density Parity-Check) codes is highly dependent on the degree distributions of the Tanner graphs which define the LDPC codes. We compare two LDPC code ensembles, one has a uniform degree distribution and the other a non-uniform one over a BEC (Binary Erasure Channel) and a BSC (Binary Symmetric Channel) thorough DE (Density Evolution). We then derive sufficient conditions on the erasure probability of a BEC and the error probability of a BSC, under which the LDPC code ensembles with uniform degree distributions outperform those with non-uniform degree distributions.