Spatially-coupled MacKay-Neal codes and Hsu-Anastasopoulos codes

Abstract

SUMMARYKudekar 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.

Spatially-coupled codes for side-information problems

“…The encoding and decoding are based on practically implementable, polynomial time message-passing algorithms. In essence, we describe new codebooks SC, SC(s k ), SC , SC (s k ), analogous to C, C(s k ), C , C (s k ), that appear to be good for rate distortion and/or channel coding when encoding and decoding is done with message-passing algorithms [13], [12], [19], [20]. Given that these are good for rate distortion and/or channel coding, the coding scheme for the side-information problems is same as in the previous section.…”

“…We refer the reader to [22] for their description. It has been recently shown [19], [20] that SC compound LDGM/LDPC codes are good for channel coding under message-passing. As such, SC, SC(s k ), SC (s k ) are good for channel coding under message-passing.…”

Finite-length scaling based on Belief Propagation for spatially coupled LDPC codes

“…Indeed, the minimum distance growth rate for the coupled LDPC ensemble is often better than for the uncoupled ensemble [5]. Several families of SC-LDPC code ensembles are compared in [5], [6] using asymptotic arguments, namely BP threshold and minimum distance growth rate.…”

Design of Irregular SC-LDPC Codes With Non-Uniform Degree Distributions by Linear Programming

“…The first approach to construct irregular SC-LDPC codes is coupling protograph-based codes [7] such as repeat-accumulate (RA) [8], accumulate-repeat-jagged-accumulate (ARJA) [9], and MacKay-Neal (MN) [10] codes. Spatially coupled ARJA codes [11], [12] and spatially-coupled MN codes [13], [14] show better asymptotic performance than regular SC-LDPC codes with bounded degrees. However, it is later known that the finite-length performance of spatiallycoupled ARJA codes is rather worse than that of other spatially-coupled codes due to their inferior scaling behavior [15].…”