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International Symposium on Turbo Codes and Iterative Information Processing

A New Algebraic-Based Method for Constructing Protograph LDPC Codes

Xijin Mu, Huaan Li, Chao Chen, Jinhong Yuan, Baoming Bai

Published Dec 1, 2018• 1 citation
Publisher

Abstract

This paper focuses on the construction of protograph low-density parity-check (PTG-LDPC) codes. We first give a necessary condition for a PTG-LDPC code to avoid 4-cycles. Then, we propose a new method for constructing PTG-LDPC codes based on the decomposition of a base matrix and the algebraic construction of an exponent matrix. The proposed method owns a lower design complexity compared with the conventional graph-based methods. Numerical results show that the proposed codes perform well, and have no error floor down to a block error rate of 10-4.

DOI
10.1109/istc.2018.8625339

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This paper focuses on the construction of protograph low-density parity-check (PTG-LDPC) codes. We first give a necessary condition for a PTG-LDPC code to avoid 4-cycles. Then, we propose a new method for constructing PTG-LDPC codes based on the decomposition of a base matrix and the algebraic construction of an exponent matrix. The proposed method owns a lower design complexity compared with the conventional graph-based methods. Numerical results show that the proposed codes perform well, and have no error floor down to a block error rate of 10-4.
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.
LDPC codes based on multi-edge protographs potentially have larger minimum distances compared to their counterparts, single-edge protographs. However, considering different features of their Tanner graph, such as short cycles, girth and other graphical structures, is harder than for Tanner graphs from single-edge protographs. Here, we provide a novel approach to construct the parity-check matrix of an LDPC code which is based on trades obtained from block designs. We employ our method to construct multi-edge quasi-cyclic (QC) LDPC codes.We use those trade-based matrices to define base matrices of multi-edge protographs. The construction of exponent matrices corresponding to these base matrices has less complexity than the ones proposed in the literature. We prove that these base matrices result in QC-LDPC codes with smaller lower bounds on the lifting degree than existing ones.
In this paper, we propose an algebraic approach to design extremely low rate low-density parity-check (LDPC) codes by employing the primitive polynomials. We first investigate the requirements for the polynomials to construct 4-cycle free LDPC codes and then propose an algebraic approach to design the parity check matrix. To achieve a better decoding performance in the extremely low rate region, we propose an extension method to construct more check nodes and virtual variable nodes to obtain more constraints. Our designed LDPC codes have simple encoding and simple decoding features. Simulation results show that our proposed rate-0.03 LDPC code can achieve 2.2 dB gain, compared to the LDPC code designed by the progressive edge growth algorithm with the optimized edge degree distributions described in [7].
A novel construction method of Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code is proposed in this paper, which is based on protograph and Balanced Block Design (BIBD) in combinational mathematics.The method is to fill the base matrix of protograph with the base set elements of BIBD in accordance with certain mathematical rules.And the base matrix of protograph proposed in the paper has special double diagonal structure, which overcomes the low weight code words in general structure and achieves rapid encoding at the same time.The QC-LDPC code in this paper possesses lower encoding complexity and less memory occupation; therefore it is easy for hardware implementation.In addition, it effectively avoids girth 4. Simulation results show that when the Bit Error Rate (BER) is 10 -7 , QC-LDPC codes based on Protograph and BIBD (PBIBD-LDPC) proposed in this paper has 0.5dB Net Coding Gain (NCG) compared with the protograph QC-LDPC codes that based on Vander monde matrix and PEG algorithm structure, and has 0.15dB NCG compared with the excellent code QC-LDPC that based on the lifting structure.
In this paper, we propose a method to design nested protograph-based low-density parity-check (LDPC) codes for network communications with low error-floors via a graph modification procedure. Lowering the error-floor of nested LDPC codes is essential for certain applications that require very low decoded error rates in moderate to high signal-to-noise-ratios (SNRs) like optical communication and data storage. The multilevel method we propose improves upon existing state-of-the-art algebraic designs by the identification and removal of harmful graph structures from nested LDPC codes, effectively improving the performance of each of the nested codes in their error-floor region. Simulation results are provided that confirm the expected performance improvement.
To reduce the cycles in the Tanner graph which may affect the code performance,a recursive method to construct low-density parity-check(LDPC) codes was proposed.The method used a parity-check matrix of a short LDPC code as its mother matrix upon which a long code was constructed with circulant permutation matrices.By choosing the circulant permutation matrix parameters properly,the number of cycles having a given length in the Tanner graph of the constructed long codes were equal to or less than that of the short code.Either regular or irregular LDPC codes were constructed.Simulation results show that the LDPC codes constructed by this method have very low error-floor and almost no performance degradation due to their special structure as compared with random LDPC codes.