Juane Li

Algebraic Quasi-Cyclic LDPC Codes: Construction, Low Error-Floor, Large Girth and a Reduced-Complexity Decoding Scheme

This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density paritycheck (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.

A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes

This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array, masking, binary to nonbinary replacement, and matrix-dispersion. By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity.

An unnoticed strong connection between algebraic-based and protograph-based LDPC codes, Part I: Binary case and interpretation

This paper unveils a strong connection between two major constructions of LDPC codes, namely the algebraic-based and the protograph-based constructions. It is shown that, from a graph-theoretic point of view, an algebraic LDPC code whose parity-check matrix is an array of submatrices of the same size over a finite field is a protograph LDPC code. Conversely, from a matrix-theoretic point of view, since the parity-check matrix of a protograph code can be arranged as an array of submatrices of the same size over a finite field and its base graph (or base matrix) can be constructed algebraically, a protograph LDPC code is an algebraic LDPC code. These two major approaches have their advantages and disadvantages in code construction. Unification of these two approaches may lead to better designs and constructions of LDPC codes to achieve good overall performance in terms of error performance in waterfall region, error-floor location and rate of decoding convergence. This paper is the first part of a series of two parts, Part-I and Part-II. Part-I investigates only the binary LDPC codes constructed by the superposition and the protograph-based methods. Part-II explores nonbinary LDPC codes from both superposition and protograph points of view. Also included in Part II are specific superposition constructions of both binary and nonbinary quasi-cyclic LDPC codes.

Decoding of quasi-cyclic LDPC codes with section-wise cyclic structure

Presented in this paper is a reduced-complexity iterative decoding scheme for quasi-cyclic (QC) LDPC codes. This decoding scheme is devised based on the section-wise cyclic structure of the parity-check matrix of a QC-LDPC code. Using this decoding scheme, the hardware implementation complexity of a QC-LDPC decoder can be significantly reduced without performance degradation. A high-rate QC-LDPC code that can achieve a very low error-rate without a visible error-floor is used to demonstrate the effectiveness of the proposed decoding scheme. Also presented in this paper are two other high-rate QC-LDPC codes and a method for constructing rate -1/2 QC-LDPC codes whose Tanner graphs have girth 8. All the codes constructed perform well with low error-floor using the proposed decoding scheme.

Improved message-passing algorithm for counting short cycles in bipartite graphs

Recently, Karimi and Banihashemi proposed an algorithm based on message-passing to count cycles in a graph of lengths less than double its girth. The algorithm uses only integer additions and subtractions to compute messages at the nodes of the graph that are passed to adjacent nodes. The complexity of the algorithm, when applied to a bipartite graph of girth g that has E edges, is O(gE2). The algorithm is superior to many other existing algorithms in the literature. In this paper, an improvement of this algorithm is presented that cuts both the complexity and the computing time by a factor of two. The improved algorithm is also applied to Tanner graphs of quasi-cyclic codes and, in this case, the complexity can be further cut by a factor of p, where p is the size of the circulants in the parity-check matrix of the quasi-cyclic code.

Quasi-cyclic LDPC codes on two arbitrary sets of a finite field

This paper presents a simple and flexible method for constructing QC-LDPC codes based on two arbitrary sets of a finite field. Based on this method, a high-rate, high-performance and very low error-floor QC-LDPC code is first constructed and then a class of rate-1/2 QC-LDPC codes whose Tanner graphs have girth 8 or larger is presented. Also presented is a reduced-complexity iterative decoding algorithm for QC-LDPC codes.

LDPC Code Designs, Constructions, and Unification

Written by leading experts, this self-contained text provides systematic coverage of LDPC codes and their construction techniques, unifying both algebraic- and graph-based approaches into a single theoretical framework (the superposition construction). An algebraic method for constructing protograph LDPC codes is described, and entirely new codes and techniques are presented. These include a new class of LDPC codes with doubly quasi-cyclic structure, as well as algebraic methods for constructing spatially and globally coupled LDPC codes. Authoritative, yet written using accessible language, this text is essential reading for electrical engineers, computer scientists and mathematicians working in communications and information theory.

Globally coupled LDPC codes

This paper presents a special type of LDPC codes with a structure related to but different from that of the spatially coupled LDPC codes. For an LDPC code of this type, its Tanner graph is composed of a set of small disjoint Tanner graphs which are connected together by a group of overall check-nodes, called global check-nodes. Codes of this type are called globally coupled LDPC codes and they perform well over both the additive Gaussian white noise and the binary-erasure channels. Furthermore, they are very effective at correcting erasures clustered in bursts. Two algebraic methods are presented for constructing these codes. A two-phase local/global iterative scheme for decoding these codes is presented. This decoding scheme allows correction of local random and global errors and/or erasures in two phases.

Construction of non-binary LDPC codes: A matrix-theoretic approach

This paper presents a matrix-theoretic approach to the construction of non-binary LDPC codes. Two algebraic methods for constructing non-binary LDPC codes are presented. The proposed construction methods have several ingredients including base matrix, matrix-dispersion, masking and replacement. By proper choice and combination of these ingredients, non-binary LDPC codes with good performance can be constructed.

A merry-go-round decoding scheme for non-binary quasi-cyclic LDPC codes

This paper presents a reduced-complexity iterative scheme and an algorithm for decoding non-binary quasi-cyclic (QC) LDPC codes of a specific type. The proposed decoding scheme and the algorithm together significantly reduce the hardware implementation complexity of a decoder with no performance degradation. Also presented in the paper is a simple method for constructing a class of non-binary QC-LDPC codes.