LDPC Codes Over Non-binary Alphabets PDF Download

Author: Ariel Amir
Publisher:
ISBN:
Category :
Languages : en
Pages : 76

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Author: Deepak Gilra
Publisher:
ISBN:
Category :
Languages : en
Pages :

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In this thesis we study Low Density Parity Check (LDPC) and LDPC like codes over non-binary fields. We extend the concepts used for non-binary LDPC codes to generalize Product Accumulate (PA) codes to non-binary fields. We present simulation results that show that PA codes over GF(4) performs considerably better than binary PA codes at smaller block lengths and slightly better at large block lengths. We also propose a trellis based decoding algorithm to decode PA codes and show that its complexity is considerably lower than the message-passing algorithm. In the second part of the thesis we study the convergence properties of non-binary PA codes and non-binary LDPC codes. We use EXIT-charts to study the convergence properties of non-binary LDPC codes with different mean column weights and show why certain irregularities are better. Although the convergence threshold predicted by EXIT-charts on non-binary LDPC codes is quite optimistic we can still use EXIT-charts for comparison between non-binary LDPC codes with different mean column weights.

Author:
Publisher: Academic Press
ISBN: 012397223X
Category : Technology & Engineering
Languages : en
Pages : 687

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This book gives a review of the principles, methods and techniques of important and emerging research topics and technologies in Channel Coding, including theory, algorithms, and applications. Edited by leading people in the field who, through their reputation, have been able to commission experts to write on a particular topic. With this reference source you will: - Quickly grasp a new area of research - Understand the underlying principles of a topic and its applications - Ascertain how a topic relates to other areas and learn of the research issues yet to be resolved - Quick tutorial reviews of important and emerging topics of research in Channel Coding - Presents core principles in Channel Coding theory and shows their applications - Reference content on core principles, technologies, algorithms and applications - Comprehensive references to journal articles and other literature on which to build further, more specific and detailed knowledge

Author: Pia Aviva Zobel
Publisher:
ISBN:
Category : Decoders (Electronics)
Languages : en
Pages : 80

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Author: Bo Zhou
Publisher:
ISBN:
Category :
Languages : en
Pages : 292

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Author: Behzad Amiri
Publisher:
ISBN:
Category :
Languages : en
Pages : 127

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We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage. One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems. In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes. In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications. In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles. The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.

Author: Amir H. Djahanshahi
Publisher:
ISBN: 9781109690071
Category :
Languages : en
Pages : 117

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Low-density parity-check (LDPC) codes have been known for their outstanding error-correction capabilities. With low-complexity decoding algorithms and a near capacity performance, these codes are among the most promising forward error correction schemes. LDPC decoding algorithms are generally sub-optimal and their performance not only depends on the codes, but also on many other factors, such as the code representation. In particular, a given non-binary code can be associated with a number of different field or ring image codes. Additionally, each LDPC code can be described with many different Tanner graphs. Each of these different images and graphs can possibly lead to a different performance when used with iterative decoding algorithms. Consequently, in this dissertation we try to find better representations, i.e., graphs and images, for LDPC codes. We take the first step by analyzing LDPC codes over multiple-input single-output (MISO) channels. In an n_T by 1 MISO system with a modulation of alphabet size 2^M, each group of n_T transmitted symbols are combined and produce one received symbol at the receiver. As a result, we consider the LDPC-coded MISO system as an LDPC code over a 2^{M n_T}-ary alphabet. We introduce a modified Tanner graph to represent MISO-LDPC systems and merge the MISO symbol detection and binary LDPC decoding steps into a single message passing decoding algorithm. We present an efficient implementation for belief propagation decoding that significantly reduces the decoding complexity. With numerical simulations, we show that belief propagation decoding over modified graphs outperforms the conventional decoding algorithm for short length LDPC codes over unknown channels. Subsequently, we continue by studying images of non-binary LDPC codes. The high complexity of belief propagation decoding has been proven to be a detrimental factor for these codes. Thereby, we suggest employing lower complexity decoding algorithms over image codes instead. We introduce three classes of binary image codes for a given non-binary code, namely: basic, mixed, and extended binary image codes. We establish upper and lower bounds on the minimum distance of these binary image codes, and present two techniques to find binary image codes with better performance under belief propagation decoding algorithm. In particular, we present a greedy algorithm to find optimized binary image codes. We then proceed by investigation of the ring image codes. Specifically, we introduce matrix-ring-image codes for a given non-binary code. We derive a belief propagation decoding algorithm for these codes, and with numerical simulations, we demonstrate that the low-complexity belief propagation decoding of optimized image codes has a performance very close to the high complexity BP decoding of the original non-binary code. Finally, in a separate study, we investigate the performance of iterative decoders over binary erasure channels. In particular, we present a novel approach to evaluate the inherent unequal error protection properties of irregular LDPC codes over binary erasure channels. Exploiting the finite length scaling methodology, that has been used to study the average bit error rate of finite-length LDPC codes, we introduce a scaling approach to approximate the bit erasure rates in the waterfall region of variable nodes with different degrees. Comparing the bit erasure rates obtained from Monte Carlo simulation with the proposed scaling approximations, we demonstrate that the scaling approach provides a close approximation for a wide range of code lengths. In view of the complexity associated with the numerical evaluation of the scaling approximation, we also derive simpler upper and lower bounds and demonstrate through numerical simulations that these bounds are very close to the scaling approximation.

Author: Ankit Ghiya
Publisher:
ISBN:
Category :
Languages : en
Pages : 48

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A family of low-density lattice codes (LDLC) is studied based on Construction-A for lattices. The family of Construction-A codes is already known to contain a large capacity-achieving subset. Parallels are drawn between coset non-binary low-density parity-check (LDPC) codes and nested low-density Construction-A lattices codes. Most of the related research in LDPC domain assumes optimal power allocation to encoded codeword. The source coding problem of mapping message to power optimal codeword for any LDPC code is in general, NP-hard. In this thesis, we present a novel method for encoding and decoding lattice based on non-binary LDPC codes using message-passing algorithms.

Author: Fang Cai
Publisher:
ISBN:
Category :
Languages : en
Pages : 149

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Non-binary low-density parity-check (NB-LDPC) codes can achieve better error-correcting performance than their binary counterparts when the code length is moderate at the cost of higher decoding complexity. The high complexity is mainly caused by the complicated computations in the check node processing and the large memory requirement. In this thesis, three decoding algorithms and corresponding VLSI architectures are proposed for NB-LDPC codes to lower the computational complexity and memory requirement. The first design is based on the proposed relaxed Min-max decoding algorithm. A novel relaxed check node processing scheme is proposed for the Min-max NB-LDPC decoding algorithm. Each finite field element of GF(2p̂) can be uniquely represented by a linear combination of $p$ independent field elements. Making use of this property, an innovative method is developed in this paper to first find a set of the p most reliable variable-to-check messages with independent field elements, called the minimum basis. Then the check-to-variable messages are efficiently computed from the minimum basis. With very small performance loss, the complexity of the check node processing can be substantially reduced using the proposed scheme. In addition, efficient VLSI architectures are developed to implement the proposed check node processing and overall NB-LDPC decoder. Compared to the most efficient prior design, the proposed decoder for a (837, 726) NB-LDPC code over GF(25̂) can achieve 52% higher efficiency in terms of throughput-over-area ratio. The second design is based on a proposed enhanced iterative hard reliability-based majority-logic decoding. The recently developed iterative hard reliability-based majority-logic NB-LDPC decoding has better performance-complexity tradeoffs than previous algorithms. Novel schemes are proposed for the iterative hard reliability-based majority-logic decoding (IHRB-MLGD). Compared to the IHRB algorithm, our enhanced (E- )IHRB algorithm can achieve significant coding gain with small hardware overhead. Then low-complexity partial-parallel NB-LDPC decoder architectures are developed based on these two algorithms. Many existing NB-LDPC code construction methods lead to quasi-cyclic or cyclic codes. Both types of codes are considered in our design. Moreover, novel schemes are developed to keep a small proportion of messages in order to reduce the memory requirement without causing noticeable performance loss. In addition, a shift-message structure is proposed by using memories concatenated with variable node units to enable efficient partial-parallel decoding for cyclic NB-LDPC codes. Compared to previous designs based on the Min-max decoding algorithm, our proposed decoders have at least tens of times lower complexity with moderate coding gain loss. The third design is based on a proposed check node decoding scheme using power representation of finite field elements. Novel schemes are proposed for the Min-max check node processing by making use of the cyclical-shift property of the power representation of finite field elements. Compared to previous designs based on the Min-max algorithm with forward-backward scheme, the proposed check node units (CNUs) do not need the complex switching network. Moreover, the multiplications of the parity check matrix entries are efficiently incorporated. For a Min-max NB-LDPC decoder over GF(32), the proposed scheme reduces the CNU area by at least 32%, and leads to higher clock frequency.

Author: John L. Fan
Publisher: Springer Science & Business Media
ISBN: 1461515254
Category : Technology & Engineering
Languages : en
Pages : 268

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Constrained Coding and Soft Iterative Decoding is the first work to combine the issues of constrained coding and soft iterative decoding (e.g., turbo and LDPC codes) from a unified point of view. Since constrained coding is widely used in magnetic and optical storage, it is necessary to use some special techniques (modified concatenation scheme or bit insertion) in order to apply soft iterative decoding. Recent breakthroughs in the design and decoding of error-control codes (ECCs) show significant potential for improving the performance of many communications systems. ECCs such as turbo codes and low-density parity check (LDPC) codes can be represented by graphs and decoded by passing probabilistic (a.k.a. `soft') messages along the edges of the graph. This message-passing algorithm yields powerful decoders whose performance can approach the theoretical limits on capacity. This exposition uses `normal graphs,' introduced by Forney, which extend in a natural manner to block diagram representations of the system and provide a simple unified framework for the decoding of ECCs, constrained codes, and channels with memory. Soft iterative decoding is illustrated by the application of turbo codes and LDPC codes to magnetic recording channels. For magnetic and optical storage, an issue arises in the use of constrained coding, which places restrictions on the sequences that can be transmitted through the channel; the use of constrained coding in combination with soft ECC decoders is addressed by the modified concatenation scheme also known as `reverse concatenation.' Moreover, a soft constraint decoder yields additional coding gain from the redundancy in the constraint, which may be of practical interest in the case of optical storage. In addition, this monograph presents several other research results (including the design of sliding-block lossless compression codes, and the decoding of array codes as LDPC codes). Constrained Coding and Soft Iterative Decoding will prove useful to students, researchers and professional engineers who are interested in understanding this new soft iterative decoding paradigm and applying it in communications and storage systems.