Mar 18, 2020

Design of LDPC Codes: A Survey and New Results Gianluigi Liva, Shumei Song, Lan Lan, Yifei Zhang, Shu Lin, and William E. Ryan

Abstract— This survey paper provides fundamentals in the design of LDPC codes. To provide a target for the code designer, we first summarize the EXIT chart technique for determining (near-)optimal degree distributions for LDPC code ensembles. We also demonstrate the simplicity of representing codes by protographs and how this naturally leads to quasi-cyclic LDPC codes. The EXIT chart technique is then extended to the special case of protograph-based LDPC codes. Next, we present several design approaches for LDPC codes which incorporate one or more accumulators, including quasi-cyclic accumulator- based codes. The second half the paper then surveys several algebraic LDPC code design techniques. First, codes based on finite geometries are discussed and then codes whose designs are based on Reed-Solomon codes are covered. The algebraic designs lead to cyclic, quasi-cyclic, and structured codes. The masking technique for converting regular quasi-cyclic LDPC codes to irregular codes is also presented. Some of these results and codes have not been presented elsewhere. The paper focuses on the binary-input AWGN channel (BI-AWGNC). However, as discussed in the paper, good BI-AWGNC codes tend to be universally good across many channels. Alternatively, the reader may treat this paper as a starting point for extensions to more advanced channels. The paper concludes with a brief discussion of open problems.

I. INTRODUCTION

The class of low-density parity-check (LDPC) codes repre- sents the leading edge in modern channel coding. They have held the attention of coding theorists and practitioners in the past decade because of their near-capacity performance on a large variety of data transmission and storage channels and because their decoders can be implemented with manageable complexity. They were invented by Gallager in his 1960 doctoral dissertation [1] and were scarcely considered in the 35 years that followed. One notable exception is Tanner, who wrote an important paper in 1981 [2] which generalized LDPC codes and introduced a graphical representation of LDPC codes, now called Tanner graphs. Apparently independent of Gallager’s work, LDPC codes were re-invented in the mid- 1990’s by MacKay, Luby, and others [3][4][5][6] who noticed the advantages of linear block codes which possess sparse (low-density) parity-check matrices.

This papers surveys the state-of-the-art in LDPC code design for binary-input channels while including a few new results as well. While it is tutorial in some aspects, it is not

Manuscript received July 04, 2006; revised August 25, 2006. This work was supported by the University of Bologna, NASA-Goddard, and NSF.

Paper has been approved by F. Chiaraluce. Gianluigi Liva is with the University of Bologna (e-mail:

gliva@deis.unibo.it). Shumei Song, Lan Lan and Shu Lin are with the University of Cal-

ifornia at Davis (e-mail: ssmsong@ece.ucdavis.edu, squashlan@gmail.com, shulin@ece.ucdavis.edu).

Yifei Zhang and William E. Ryan are with the University of Arizona, U.S.A. (e-mail: {shulin, ryan}@ece.arizona.edu).

entirely a tutorial paper, and the reader is expected to be fairly versed on the topic of LDPC codes. Tutorial coverages of LDPC codes can be found in [7][8]. The purpose of this paper is to give the reader a detailed overview of various LDPC code design approaches and also to point the reader to the literature. While our emphasis is on code design for the binary-input AWGN channel (BI-AWGNC), the results in [9][10][11][12] demonstrate that a LDPC code that is good on the BI-AWGNC tends to be universally good and can be expected to be good on most wireless, optical, and storage channels.

We favor code designs which are most appropriate for appli- cations, by which we mean codes which have low-complexity encoding, good waterfall regions, and low error floors. Thus, we discuss quasi-cyclic (QC) codes because their encoders may be implemented by shift-register circuits [13]. We also discuss accumulator-based codes because low-complexity en- coding is possible from their parity-check matrices, whether they are quasi-cyclic or not. The code classes discussed tend to be the ones (or related to the ones) used in applications or adopted for standards. Due to time and space limitations, we cannot provide a complete survey. The present survey is biased toward the expertise and interests of the authors.

Before a code can be designed, the code designer needs to know the design target. For this reason, Section II first briefly reviews the belief propagation decoder for LDPC codes and then presents the so-called extrinsic information transfer (EXIT) chart technique for this decoder. The EXIT chart technique allows one to obtain near-optimal parameters for LDPC code ensembles which guide the code designer. The EXIT technique is extended in Section III to the case of codes based on protographs. Section IV considers LDPC codes based on accumulators. The code types treated in that section are: repeat-accumulate, irregular repeat-accumulate, irregular repeat-accumulate-accumulate, generalized irregular repeat-accumulate, and accumulate-repeat-accumulate. That section also gives examples of quasi-cyclic code design using protograph (or base matrix) representations. Section V surveys the literature on cyclic and quasi-cyclic LDPC code design based on finite geometries. Section VI presents several LDPC code design techniques based on Reed-Solomon codes. Section VII presents the masking technique for converting regular QC codes to irregular QC codes to conform to prescribed code parameters. Section VIII contains some concluding remarks and some open problems.

II. DESIGN VIA EXIT CHARTS

We start with an m × n low-density parity-check matrix H, which corresponds to a code with design rate (n−m)/n, which could be less than the actual rate, R = k/n, where k is the number of information bits per codeword. H gives rise

Fig. 1. Tanner graph representation of LDPC codes.

to a Tanner graph which has m check nodes, one for each row of H, and n variable nodes, one for each column of H. Considering the general case in which H has non-uniform row and column weight, the Tanner graph can be characterized by degree assignments {dv(i)}ni=1 and {dc(j)}mj=1, where dv(i) is the degree of the i-th variable node and dc(j) is the degree of the j-th check node. Such a graph, depicted in Fig. 1, is representative of the iterative decoder, with each node representing a soft-in/soft-out processor (or node decoder).

We shall assume the BI-AWGNC in our description of the LDPC iterative decoder. In this model, a received channel sample y is given by y = x + w, where x = (−1)c ∈ {±1} is the bipolar representation of the transmitted code bit c ∈ {0, 1} and w is a white Gaussian noise sample distributed as η

( 0, σ2w

) , where σ2w = N0/2, following convention. The

channel bit log-likelihood ratios (LLRs) are computed as

Lch = log (

p (x = +1 | y) p (x = −1 | y)

) =

2y σ2w

. (1)

In one iteration of the conventional, flooding-schedule iter- ative decoder, the variable node decoders (VNDs) first process their input LLRs and send the computed outputs (messages) to each of their neighboring check node decoders (CNDs); then the CNDs process their input LLRs and send the computed outputs (messages) to each of their neighboring VNDs. More specifically, the message from the i-th VND to the j-th CND is

Li→j = Lch,i + ∑

j′ 6=j Lj′→i (2)

where Lj′→i is the incoming message from CND j′ to VND i and where the summation is over the dv(i)− 1 check node neighbors of variable node i, excluding check node j. The message from CND j to VND i is given by

Lj→i = 2 tanh−1 (

∏ i′ 6=i

tanh (Li′→j)

) (3)

where Li′→j is the incoming message from VND i′ to CND j and where the product is over the dc(j) − 1 variable node neighbors of check node j, excluding variable node i . This decoding algorithm is called the sum-product algorithm (SPA).

We now discuss the EXIT chart technique [14][15][11] for this decoder and channel model. The idea is that the VNDs and the CNDs work cooperatively and iteratively to make bit decisions, with the metric of interest generally improving with each half-iteration. A transfer curve which plots the input metric versus the output metric can be obtained for both the VNDs and the CNDs, where the transfer curve for the VNDs depends on the channel SNR. Further, since the output metric for one processor is the input metric for its companion processor, one can plot both transfer curves on the same axes, but with the abscissa and ordinate reversed for one processor. Such a chart aids in the prediction of the decoding threshold of the ensemble of codes characterized by given VN and CN degree distributions: the decoding threshold is the SNR at which the two transfer curves just touch, precluding convergence of the two processors. EXIT chart computations are thus integral to the optimization of Tanner graph node degree distributions for LDPC codes and are the main computation in the optimization process. We emphasize that decoding threshold prediction techniques such as EXIT charts or density evolution [16] assume a graph with no cycles, an infinite codeword length, and an infinite number of decoding iterations.

An EXIT chart example is depicted in Fig. 2 for the ensemble of regular LDPC codes on the BI-AWGNC with dv(i) = dv = 3 for i = 1, ..., n, and dc(j) = dc = 6 for j = 1, ..., m. In the figure, the metric used for the transfer curves is extrinsic mutual informa

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