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Spatially-coupled binary MacKay-Neal codes for channels with non-binary inputs and affine subspace outputs

健太 笠井, Kenta Kasai, 隆之 野崎, Takayuki Nozaki, 好一 坂庭, KOHICHI SAKANIWA

Published Jun 1, 2012• 0 citations
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Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially-coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes. It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes achieve the capacity of memoryless binary-input symmetric-output channels under MAP decoding with bounded column and row weight of the parity-check matrices. The HA codes and the MN codes are dual codes each other. The aim of this paper is to present an empirical evidence that spatially-coupled MN (resp. HA) codes with bounded column and row weight achieve the capacity of the BEC. To this end, we introduce a spatial coupling scheme of MN (resp. HA) codes. By density evolution analysis, we will show that the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close to the Shannon limit.

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Asymptotic analysis of spatially coupled MacKay-Neal and Hsu-Anastasopoulos LDPC codes

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MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) low-density parity-check (LDPC) codes are known to achieve the capacity of memoryless binary-input symmetric-output channels under maximum likelihood (ML) decoding with bounded col- umn and row weight in their associated parity-check matrices. Recently, Kasai and Sakaniwa showed that spatially coupled (SC) versions of the MN and HA LDPC codes have belief propagation (BP) iterative decoding thresholds that approach capacity on the binary erasure channel (BEC) as the coupling length increases. In this paper, we extend the results of Kasai and Sakaniwa to the additive white Gaussian noise (AWGN) channel and show that the thresholds of the SC-MN and SC-HA ensembles approach capacity with bounded density as the coupling length increases, i.e., the number of edges per information bit approaches a finite value as the estimated BP threshold approaches the Shannon limit. We also perform an asymptotic weight enumerator analysis and show that, provided the density parameters are chosen to be sufficiently large, the SC-MN and SC-HA ensembles are asymptotically good. Further, for certain selections of parameters, some of these ensembles are shown to have both excellent thresholds and good distance properties.

International Symposium on Information Theory and its Applications•Oct 1, 2012•5 citations
We study LDPC codes for the channel with 2m-ary input x ϵ F2mand output y = x + z ϵ F2m. The receiver knows a subspace V ⊂ F2mfrom which z = y - x is uniformly chosen. Or equivalently, the receiver receives an affine subspace y-V where x lies. We consider a joint iterative decoder involving the channel detector and the LDPC decoder. The decoding system considered in this paper can be viewed as a simplified model of the joint iterative decoder over non-binary modulated signal inputs e.g., 2m-QAM. We evaluate the performance of binary spatially-coupled MacKay-Neal codes by density evolution. The iterative decoding threshold is seriously degraded by increasing m. EXIT-like function curve calculations reveal that this degradation is caused by wiggles and can be mitigated by increasing the randomized window size. The resultant iterative decoding threshold values are very close to the Shannon limit.
Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially-coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes. It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes achieve the capacity of memoryless binary-input symmetric-output channels under MAP decoding with bounded column and row weight of the parity-check matrices. The HA codes and the MN codes are dual codes each other. The aim of this paper is to present an empirical evidence that spatially-coupled MN (resp. HA) codes with bounded column and row weight achieve the capacity of the BEC. To this end, we introduce a spatial coupling scheme of MN (resp. HA) codes. By density evolution analysis, we will show that the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close to the Shannon limit.
MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) low-density parity-check (LDPC) codes are known to achieve the capacity of memoryless binary-input symmetric-output channels under maximum likelihood (ML) decoding with bounded col- umn and row weight in their associated parity-check matrices. Recently, Kasai and Sakaniwa showed that spatially coupled (SC) versions of the MN and HA LDPC codes have belief propagation (BP) iterative decoding thresholds that approach capacity on the binary erasure channel (BEC) as the coupling length increases. In this paper, we extend the results of Kasai and Sakaniwa to the additive white Gaussian noise (AWGN) channel and show that the thresholds of the SC-MN and SC-HA ensembles approach capacity with bounded density as the coupling length increases, i.e., the number of edges per information bit approaches a finite value as the estimated BP threshold approaches the Shannon limit. We also perform an asymptotic weight enumerator analysis and show that, provided the density parameters are chosen to be sufficiently large, the SC-MN and SC-HA ensembles are asymptotically good. Further, for certain selections of parameters, some of these ensembles are shown to have both excellent thresholds and good distance properties.