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.
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.
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. (Less)
Convolutional (or spatially-coupled) low-density parity-check (LDPC) codes have now been shown to approach capacity for a variety of problems. Yet, most of these results require sequences of regular LDPC ensembles with increasing variable and check degrees. Previously, Kasai and Sakaniwa showed empirically that, for the BEC, this limitation can be overcome by using spatially-coupled MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) ensembles. In this paper, we prove this analytically for (k, 2, 2)-MN and (2, k, 2)-HA ensembles when k is at least 3. The proof is based on the simple approach to threshold saturation, introduced by Yedla et al., which relies on potential functions. The key step is verifying the non-negativity of a potential function associated with the uncoupled system. Along the way, we derive the potential function general multi-edge type (MET) LDPC ensembles and establish a duality relationship between dual ensembles of MET LDPC codes.
Low-density parity-check (LDPC) convolutional codes have been shown to exhibit excellent performance under low-complexity belief-propagation decoding [1], [2]. This phenomenon is now termed threshold saturation via spatial coupling. The underlying principle behind this appears to be very general and spatially-coupled (SC) codes have been successfully applied in numerous areas. Recently, SC regular LDPC codes have been proven to achieve capacity universally, over the class of binary memoryless symmetric (BMS) channels, under belief-propagation decoding [3], [4]. In [5], [6], potential functions are used to prove that the BP threshold of SC irregular LDPC ensembles saturates, for the binary erasure channel, to the conjectured MAP threshold (known as the Maxwell threshold) of the underlying irregular ensembles. In this paper, that proof technique is generalized to BMS channels, thereby extending some results of [4] to irregular LDPC ensembles. We also believe that this approach can be expanded to cover a wide class of graphical models whose message-passing rules are associated with a Bethe free energy.
Low-density parity-check (LDPC) convolutional codes have been shown to exhibit excellent performance under low-complexity belief-propagation decoding [1], [2]. This phenomenon is now termed threshold saturation via spatial coupling. The underlying principle behind this appears to be very general and spatially-coupled (SC) codes have been successfully applied in numerous areas. Recently, SC regular LDPC codes have been proven to achieve capacity universally, over the class of binary memoryless symmetric (BMS) channels, under belief-propagation decoding [3], [4]. In [5], [6], potential functions are used to prove that the BP threshold of SC irregular LDPC ensembles saturates, for the binary erasure channel, to the conjectured MAP threshold (known as the Maxwell threshold) of the underlying irregular ensembles. In this paper, that proof technique is generalized to BMS channels, thereby extending some results of [4] to irregular LDPC ensembles. We also believe that this approach can be expanded to cover a wide class of graphical models whose message-passing rules are associated with a Bethe free energy.
For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check “should be large”: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices “should not be large”: The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum “non-coupled” quasi-cyclic LDPC codes.