2022-11-12 07:11

Reliable and efficient transmission/storage is one of the ultimate design objectives of modern communication/data storage systems. Among the existing techniques, forward error correction (FEC) is a powerful solution to enhance the reliability of data transmission and storage. The idea of FEC is that incorporating some redundancy into the information bits so as to form a codeword. Instead of the original information, the codeword is transmitted to the receiver through a channel. As a result, the information bits can be successfully retrieved if the codeword is appropriately designed [1].

The history of FEC codes can be dated back to six decades ago. In 1948, C. E. Shannon proved that the error-free (EF) transmission can be achieved for every noisy channel by means of FEC codes with any code rate up to the channel capacity (i.e., maximum achievable code rate) [2], which is called as Shannon theory. Thereby, the maximum amount of the incorporated redundancy that is utilized to ensure correct decoding is determined by the channel capacity. Unfortunately, this theory just quantifies the channel capacity, but does not elaborate how to construct the capacity-approaching codes. Since then, many researchers have endeavored to search the capacityapproaching codes but obtained slow progress. The first FEC code, namely Hamming code, was invented in 1950 [3]. Later, a new FEC code, namely low-density parity-check (LDPC) code, was introduced by R. G. Gallager in his doctoral dissertation [4]. However, in the following 35 years, LDPC codes and their variants were nearly neglected and scarcely considered. One notable contribution during that period is the work of R. M. Tanner [5], who has proposed a graphical representation of the LDPC code—Tanner graph.

In 1993, the turbo code, which enables error performance approaching the channel capacity over additive white Gaussian noise (AWGN) channels, was invented by Berrou et al. [6]. In the wake of the remarkable success of turbo codes, the LDPC codes have been rediscovered by Mackay et al. because of their capacity-approaching performance under the iterative decoding algorithm [7]–[11]. As compared with turbo codes, LDPC codes possess two distinct advantages: low error-floor and fast decoding [11]–[13]. However, the relatively high encoding complexity is one practical weakness of the LDPC codes. To tackle this problem, several methods have been proposed [14]–[16]. Owing to the above-mentionedadvantages, a significant amount of effort has been devoted to analyzing and designing the LDPC codes [12]–[29]. In particular, the density evolution (DE) [12], [13] and extrinsic information transfer (EXIT) function [17]–[19], [29] have been considered as two most popular techniques for optimization of LDPC codes. Besides, the iterative decoding algorithm of LDPC codes has been carefully improved so as to obtain a better decoding performance [30]–[33]. Nowadays, LDPC codes have been widely applied in a myriad of communication and data storage systems, e.g., deep-space communication systems [34], [35], wireless communication systems [36], optical communication systems [37], and magnetic recoding systems [38], [39], and have become one of the most intensely investigated areas in channel coding. The tutorial-style articles of LDPC codes can be found in [26], [40]–[44].

Nevertheless, most capacity-approaching LDPC codes are irregular and hence suffer from the drawback of quadratic encoding complexity. Aiming at overcoming this drawback, one type of structured LDPC codes, namely quasi-cyclic (QC) LDPC codes, has been proposed [45]. Besides, a novel class of LDPC code, namely multi-edge-type (MET) LDPC codes, has been introduced by Richardson et al. [42], [46]. As a subclass of MET-LDPC codes, the protograph LDPC codes have emerged as a promising coding scheme because of their excellent error performance and low complexities [47]–[52]. Two classical types of protograph codes, namely accumulaterepeat-3-accumulate (AR3A) code and accumulate-repeat-by- 4-jagged-accumulate (AR4JA) code, which can realize linear encoding complexity and fast decoding, have been proposed by Jet Propulsion Laboratory [48]–[50]. Employing the concept of generalized LDPC codes [53], other block code constraints such as Hamming codes and recursive systematic convolutional (RSC) codes can be easily inserted into the protographs to replace selected single-parity-check (SPC) nodes (i.e., check nodes (CNs)) and to form powerful generalized protograph LDPC codes (i.e., doped-Tanner codes) [52], [54]–[57]. In order to facilitate the analysis and design of protograph LDPC codes, the protograph EXIT (PEXIT) algorithm and asymptotic weight enumerator have been proposed in [28], [58] and [50]–[52], respectively. Based on the aforementioned analytical tools, protograph LDPC codes that both possess the capacityapproaching decoding thresholds and linear-minimum-distance property have been constructed [48], [50].

Moreover, the rate-compatible protograph (RCP) LDPC codes have been studied for applications of adaptive coding and hybrid automatic repeat request (HARQ) [50], [59]–[62]. Since non-binary LDPC codes not only can outperform their binary counterparts in some cases (e.g., for short/moderate codeword length), but also can be seamlessly combined with high-order modulations [37], [63]–[65], protograph LDPC codes have been extended to the non-binary domain recently [66]–[69]. As a further advancement, the protograph framework has been introduced for convolutional codes [70] so as to form the protograph LDPC convolutional codes [71]–[73], which have certain advantages as compared to protograph LDPC codes without increasing complexity. In addition, some other variants of protograph codes, such as protograph-based raptor-like (PBRL) codes and spatially-coupled protograph-based (SCPB) codes, have been developed [74]–[78]. Aiming at further reducing the implementation complexity, QC-protograp


可靠和高效的传输/存储是现代通信/数据存储系统的最终设计目标之一。在现有技术中,前向纠错(FEC)是增强数据传输和存储可靠性的强大解决方案。 FEC的思想是将一些冗余结合到信息比特中以形成码字。代码字不是原始信息,而是通过信道发送到接收器。因此,如果码字被适当设计,则可以成功检索信息比特[1]。

FEC代码的历史可以追溯到六十年前。1948年,CE Shannon证明了通过FEC码可以实现每个噪声信道的无差错(EF)传输,任何码率都高达信道容量(即最大可实现的码率)[2],这是称为香农理论。因此,用于确保正确解码的并入冗余的最大量由信道容量确定。不幸的是,这个理论只是量化了信道容量,但没有详细说明如何构建容量接近代码。从那以后,许多研究人员一直在努力寻找容量接近代码,但进展缓慢。第一个FEC代码,即汉明码,是1950年发明的[3]。后来,R.G.Gallager在他的博士论文中引入了一种新的FEC代码,即低密度奇偶校验(LDPC)代码[4]。然而,在接下来的35年中,LDPC代码及其变体几乎被忽视,几乎没有考虑过。在此期间的一个值得注意的贡献是R. M. Tanner [5]的工作,他提出了LDPC码 - Tanner图的图形表示。

在1993年,Berrou等人发明了turbo码,它使误码性能接近加性高斯白噪声(AWGN)信道的信道容量。 [6]。随着turbo码的显着成功,Mackay等人重新发现了LDPC码。因为它们在迭代解码算法下的容量接近性能[7]-[11]。与turbo码相比,LDPC码具有两个明显的优点:低错误率和快速解码[11] -[13]。然而,相对高的编码复杂度是LDPC码的一个实际弱点。为了解决这个问题,已经提出了几种方法[14]-[16]。由于上述优点,已经花费了大量精力来分析和设计LDPC码[12]-[29]。特别地,密度演化(DE)[12],[13]和外在信息传递(EXIT)函数[17]-[19],[29]被认为是用于优化LDPC码的两种最流行的技术。此外,LDPC码的迭代译码算法已经过仔细改进,以获得更好的解码性能[30]-[33]。如今,LDPC码已广泛应用于无数通信和数据存储系统,例如深空通信系统[34],[35],无线通信系统[36],光通信系统[37]和磁记录。系统[38],[39],已成为信道编码中研究最为激烈的领域之一。LDPC代码的教程式文章可以在[26],[40]-[44]中找到。


此外,速率兼容的原型(RCP)LDPC码已经被研究用于自适应编码和混合自动重复请求(HARQ)的应用[50],[59]-[62]。由于非二进制LDPC码在某些情况下不仅可以胜过其二进制对应物(例如,对于短/中等码字长度),而且还可以与高阶调制无缝结合[37],[63]-[65],原型LDPC码最近已扩展到非二进制域[66]-[69]。作为进一步的进步,已经为卷积码引入了原型框架[70],以便形成原始图LDPC卷积码[71]-[73],与原始LDPC码相比具有某些优点而不增加复杂性。此外,还开发了一些原型机代码的其他变体,例如基于原型的猛禽(PBRL)代码和基于空间耦合的原型(SCPB)代码[74]-[78]。为了进一步降低实现复杂度,已经通过使用循环置换规则[79],[80]提出了QC-protograph LDPC码。受上述优势的启发,许多研究团体已将注意力转向原始LDPC码,因此它们似乎是不久的将来FEC码的新主要方向。





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