Turbo and LDPC (low-density parity check) codes are simple and new types of error correction codes which give a powerful and practical performance of error correction. Although experimental results show their efficacy, further theoretical analysis is necessary, which is not straightforward. We have built a unified framework of turbo and LDPC codes based on information geometry. The framework helps our intuitive understanding of the codes and opens a new prospect of further analysis. We have revealed some properties of these codes in the proposed framework. This paper summarizes the results.
Turbo and LDPC (low-density parity check) codes are simple and new types of error correction codes which give a powerful and practical performance of error correction. Although experimental results show their efficacy, further theoretical analysis is necessary, which is not straightforward. We have built a unified framework of turbo and LDPC codes based on information geometry. The framework helps our intuitive understanding of the codes and opens a new prospect of further analysis. We have revealed some properties of these codes in the proposed framework. This paper summarizes the results.
Since the proposal of turbo codes in 1993, many studies have appeared on this simple and new type of codes which give a powerful and practical performance of error correction. Although experimental results strongly support the efficacy of turbo codes, further theoretical analysis is necessary, which is not straightforward. It is pointed out that the iterative decoding algorithm of turbo codes shares essentially similar ideas with low-density parity-check (LDPC) codes, with Pearl's belief propagation algorithm applied to a cyclic belief diagram, and with the Bethe approximation in statistical physics. Therefore, the analysis of the turbo decoding algorithm will reveal the mystery of those similar iterative methods. In this paper, we recapture and extend the geometrical framework initiated by Richardson to the information geometrical framework of dual affine connections, focusing on both of the turbo and LDPC decoding algorithms. The framework helps our intuitive understanding of the algorithms and opens a new prospect of further analysis. We reveal some properties of these codes in the proposed framework, including the stability and error analysis. Based on the error analysis, we finally propose a correction term for improving the approximation.
The focus of this paper is directed towards the investigation of the characteristics of symbol-by-symbol iterative decoding algorithms for error-correcting block product-codes (block turbo-codes) which enable to reliable information transfer at relatively low received signal/noise and provide high power efficiency. Specific feature of investigated product codes is construction with usage of low-density parity-check codes (LDPC) and these code constructions are in the class of LDPC too. According to this fact the considered code constructions have symbol-by-symbol decoding algorithms developed for total class LDPC codes, namely BP (belief propagation) and its modification MIN_SUM_BP. The BP decoding algorithm is iterative and for implementation the signal/noise is required, for implementation of MIN_SUM_BP decoding algorithm the signal/noise is not required. The resulted characteristics of product codes constructed with usage of LDPC based on project geometry (length of code words, information volume, code rate, error performances) are presented in this paper. These component LDPC codes are cyclic and have encoding and decoding algorithms with low complexity implementation. The computer simulations for encoding and iterative symbol-by-symbol decoding algorithms for the number of turbo-codes with different code rate and information volumes are performed. The results of computer simulations have shown that MIN_SUM_BP decoding algorithm is more effective than BP decoding algorithm for channel with additive white gaussian noise concerning error-performances.
Low density parity check (LDPC) codes with iterative decoding based on belief propagation (IDBP) achieve astonishing error performance close to the Shannon limit. Until now there has been no known method for constructing these Shannon limit approaching codes systematically. Good LDPC codes are largely generated by computer search. As a result, the encoding of long LDPC codes is in general very complex. This paper presents the first algebraic method for constructing LDPC codes systematically based on finite analytic geometries. Four classes of finite geometry LDPC codes with relatively good minimum distances are constructed. These codes are either cyclic or quasi-cyclic and therefore their encoding can be implemented with simple linear feedback shift registers. Long finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.
The discovery of turbo codes [5] and the subsequent rediscovery of low-density parity-check (LDPC) codes [9, 18] represent a major milestone in the field of coding theory. These two classes of codes can achieve realistic bit error rates, between 10−5 and 10−12, with signalto- noise ratios that are only slightly above the minimum possible for a given channel and code rate established by Shannon’s original capacity theorems. In this sense, these codes are said to be near-capacity-achieving codes and are sometimes considered to have solved (in the engineering sense, at least) the coding problem for the additive white Gaussian noise (AWGN) channel and its derivative channels. Perhaps the most important commonality between turbo and low-density parity-check codes is that they both utilize iterative message-passing decoding algorithms. For turbo codes, one uses the so-called turbo decoding algorithm, and for LDPC codes, both the sumproduct (SP) and the min-sum (MS) algorithms are used. The success of the various iterative message-passing algorithms is sometimes said to have ushered in a new era of “modern” coding theory in which the design emphasis has shifted from optimizing some code property, such as minimum distance, to optimizing the corresponding decoding structure of the code, such as the degree profile [24, 25], with respect to the behavior of a message-passing decoder. As successful as these codes and decoders have been in terms of application, there are several major questions that must be answered before a complete understanding of them can be achieved. The theoretical research in the area of capacity-achieving codes is focused on two main themes. The first theme is whether different types of capacity-achieving codes have common encoder and structural properties. In [17], it was claimed that turbo codes could be viewed as LDPC codes, but the relationship was not made explicit. More recently, P´erez, his student Jiang, and others [11, 16] developed a construction for the parity-check matrices of arbitrary turbo codes that clearly connects the components of the turbo encoder to the resulting structure of the parity-check matrix. From a more abstract perspective, turbo codes and low-density parity-check codes are examples of codes with long block lengths that exhibit the random structure inherent in Shannon’s original theorems. The second and more active research theme is the determination of the behavior of iterative message-passing decoding and the relationships between the various decoding algorithms. The dominant problem in this area is to understand the non-codeword decoder errors that occur in computer simulations of LDPC codes with iterative message-passing decoders. Motivated by empirical observations of the non-codeword outputs of LDPC decoders, the notion of stopping sets was first introduced by Forney, et al. [8] in 2001. Two years later, a formal definition of stopping sets was given by Changyan, et al. [6]. They demonstrated that the bit and block error probabilities of iteratively decoded LDPC codes on the binary erasure channel (BEC) can be determined exactly from the stopping sets of the parity-check matrix. (Here, a stopping set S is a subset of the set of variable nodes such that all neighboring check nodes of S are connected to S at least twice.)
Abstract Recent breakthroughs in forward error correction, in the form of low-density parity-check(LDPC) and turbo codes, have seen the promise of near Shannon limit information trans-mission realised. The deterministic construction of good codes, for LDPC codes in par-ticular, is currently the subject of intense interest in the research and development com-munity. This thesis adds to this field, developing methods and supporting theory for thedesign of algebraic LDPC codes using incidence structures from combinatorial designs.The benefits of this approach are guaranteed code properties, parameter flexibility, easeof implementation and reduced encoding complexity.In the first part of this thesis Steiner 2-designs and partial geometries are used toprovide systematic constructions of regular LDPC codes with Tanner graphs free of 4-cyclesandwithdeterministiccodeproperties. Drawingextensivelyfromexistingincidencestructures we present new families of algebraic LDPC codes with benefits over traditionalrandomly constructed regular codes, particularly where small-to-medium length codes areconcerned. We focus in particular on exploring the effect of different design propertieson the code parameters by deriving the rate, minimum distance, girth, and stopping setdistribution of LDPC codes from designs.In the second part of this thesis resolvable designs are employed to create regularLDPC codes free of 4-cycles with the parameter flexibility and decoding performance ofrandomly constructed regular LDPC codes. Employing resolvability greatly extends theclass of LDPC codes that can be algebraically constructed. These codes can offer sig-nificant implementation advantages due to the deterministic structure of their resolutionclasses and in some cases their simple encoding.A shortcoming of LDPC codes is their potentially high encoding complexity, whichis generally quadratic in the code length. Finding computationally efficient encodersis therefore critical for LDPC codes to be considered as serious contenders for futuregenerations of forward error correction devices. As a result the purpose of the last part ofthis thesis is to design codes which have the graph-based properties necessary to performwell with sum-product decoding but which are also cyclic or quasi-cyclic and so can beencoded simply. The main focus is regular codes but we also extend our constructionof quasi-cyclic codes to produce irregular LDPC codes with simple linear-time encodingcircuits.