Hierarchical and High-Girth QC LDPC Codes

We present a general approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation. Our methodology starts by defining a new class of "hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. We show how to map any class of codes defined using a protograph into a family of HQC LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with designed examples of girth-10 QC LDPC codes obtained from protographs of one-sided spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor

Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.

Sub-graph based joint sparse graph for sparse code multiple access systems

Sparse code multiple access (SCMA) is a promising air interface candidate technique for next generation mobile networks, especially for massive machine type communications (mMTC). In this paper, we design a LDPC coded SCMA detector by combining the sparse graphs of LDPC and SCMA into one joint sparse graph (JSG). In our proposed scheme, SCMA sparse graph (SSG) defined by small size indicator matrix is utilized to construct the JSG, which is termed as sub-graph based joint sparse graph of SCMA (SG-JSG-SCMA). In this paper, we first study the binary-LDPC (B-LDPC) coded SGJSG- SCMA system. To combine the SCMA variable node (SVN) and LDPC variable node (LVN) into one joint variable node (JVN), a non-binary LDPC (NB-LDPC) coded SG-JSG-SCMA is also proposed. Furthermore, to reduce the complexity of NBLDPC coded SG-JSG-SCMA, a joint trellis representation (JTR) is introduced to represent the search space of NB-LDPC coded SG-JSG-SCMA. Based on JTR, a low complexity joint trellis based detection and decoding (JTDD) algorithm is proposed to reduce the computational complexity of NB-LDPC coded SGJSG- SCMA system. According to the simulation results, SG-JSGSCMA brings significant performance improvement compare to the conventional receiver using the disjoint approach, and it can also outperform a Turbo-structured receiver with comparable complexity. Moreover, the joint approach also has advantages in terms of processing latency compare to the Turbo approaches

Nonbinary Spatially-Coupled LDPC Codes on the Binary Erasure Channel

We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes, hence we conjecture that the BP threshold for large termination length approaches the MAP threshold of the underlying regular ensemble.Comment: Submitted to IEEE International Conference on Communications 201