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Cambridge University Press eBooks

Definitions, Concepts, and Fundamental Characteristics of LDPC Codes

Juane Li, Shu Lin, Khaled Abdel-Ghaffar, W.E. Ryan, Daniel J. Costello

Published Dec 1, 2016• 0 citations
Publisher

Abstract

In this chapter, we first present some definitions and concepts of matrices which are relevant for later developments of various algebraic-based and graph-theoretic-based constructions of LDPC codes. Then, we give a brief presentation of LDPC codes and characterize their fundamental structural properties that affect their error performance over the AWGNC and the BEC. Good coverage of the fundamentals of LDPC codes and the major iterative algorithms for decoding these codes can be found in [74, 97, 95]. Matrices and Matrix Dispersions of Finite Field Elements Let GF( q ) be a finite field with q elements, where q is a power of a prime [73]. GF( q ) is called a NB field if q ≠ 2 and a binary field if q = 2. A matrix A is said to be regular if every row has the same weight, called the row weight, every column has the same weight, called the column weight, and the row weight equals the column weight. If the row weight (or column weight) of a regular matrix A is w , we say that A has weight w and call it a weight- w regular matrix. For w ≠ 0, a weight- w regular matrix must be a square matrix. A binary weight-1 regular matrix is a permutation matrix (PM). It is clear that a weight- w regular matrix is the sum of w disjoint PMs. More generally, a rectangular matrix with constant column weight w c and constant row weight w r is said to be ( w c , w r ) -regular. A square matrix over GF( q ) is called a circulant if every row of the matrix is the cyclic-shift of the row above it one place to the right, and the top row is the cyclic-shift of the last row one place to the right. It is clear that a circulant is a regular matrix whose weight equals the weight of its top row. The top row of a circulant is called the generator of the circulant. A binary circulant of weight 1 is a binary circulant permutation matrix (CPM). It is clear that a weight- w circulant is the sum of w disjoint CPMs.

DOI
10.1017/9781316780152.003

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Low-density parity-check (LDPC) codes have been shown to have good error correcting performance approaching Shannon’s limit. Good error correcting performance enables efficient and reliable communication. However, a LDPC code decoding algorithm needs to be executed efficiently to meet cost, time, power and bandwidth requirements of target applications. The constructed codes should also meet error rate performance requirements of those applications. Since their rediscovery, there has been much research work on LDPC code construction and implementation. LDPC codes can be designed over a wide space with parameters such as girth, rate and length. There is no unique method of constructing LDPC codes. Existing construction methods are limited in some way in producing good error correcting performing and easily implementable codes for a given rate and length. 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•Jan 1, 2007•24 citations
In this chapter, we first present some definitions and concepts of matrices which are relevant for later developments of various algebraic-based and graph-theoretic-based constructions of LDPC codes. Then, we give a brief presentation of LDPC codes and characterize their fundamental structural properties that affect their error performance over the AWGNC and the BEC. Good coverage of the fundamentals of LDPC codes and the major iterative algorithms for decoding these codes can be found in [74, 97, 95]. Matrices and Matrix Dispersions of Finite Field Elements Let GF( q ) be a finite field with q elements, where q is a power of a prime [73]. GF( q ) is called a NB field if q ≠ 2 and a binary field if q = 2. A matrix A is said to be regular if every row has the same weight, called the row weight, every column has the same weight, called the column weight, and the row weight equals the column weight. If the row weight (or column weight) of a regular matrix A is w , we say that A has weight w and call it a weight- w regular matrix. For w ≠ 0, a weight- w regular matrix must be a square matrix. A binary weight-1 regular matrix is a permutation matrix (PM). It is clear that a weight- w regular matrix is the sum of w disjoint PMs. More generally, a rectangular matrix with constant column weight w c and constant row weight w r is said to be ( w c , w r ) -regular. A square matrix over GF( q ) is called a circulant if every row of the matrix is the cyclic-shift of the row above it one place to the right, and the top row is the cyclic-shift of the last row one place to the right. It is clear that a circulant is a regular matrix whose weight equals the weight of its top row. The top row of a circulant is called the generator of the circulant. A binary circulant of weight 1 is a binary circulant permutation matrix (CPM). It is clear that a weight- w circulant is the sum of w disjoint CPMs.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and quasi-cyclic LDPC codes with large minimum weights are shown to have no harmful trapping sets with size smaller than their minimum weights. Consequently, their performance error-floors are dominated by their minimum weights.
We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most (J+1)!(J+1)!(J+1)!, where JJJ is the column weight. For every fixed pair of column and row weights $2\le J
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.
An approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is presented. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is used to characterize certain structural properties of low-density parity-check (LDPC) codes such as the girths of their Tanner graphs. Many constructions of quasi-cyclic LDPC codes can be unified under the proposed approach.
Introduction The construction of binary low-density parity-check (LDPC) codes simply involves replacing a small number of the values in an all-zeros matrix by 1s in such a way that the rows and columns have the required degree distribution. In many cases, randomly allocating the entries in H will produce a reasonable LDPC code. However, the construction of H can affect the performance of the sum–product decoder, significantly so for some codes, and also the implementation complexity of the code. While there is no one recipe for a “good” LDPC code, there are a number of principles that inform the code designer. The first obvious decisions are which degree distribution to choose and how to construct the matrix with the chosen degrees, i.e. pseudo-randomly or with some sort of structure. Whichever construction is chosen, the features to consider include the girth of the Tanner graph and the minimum distance of the code. In this chapter we will discuss those properties of an LDPC code that affect its iterative decoding performance and then present the common construction methods used to produce codes with the preferred properties. Following common practice in the field we will call the selection of the degree distributions for an LDPC code code design and the methods to assign the locations in the parity-check matrix for the 1 entries code construction .
Low-density parity-check (LDPC) codes have been shown to have good error correcting performance approaching Shannon’s limit. Good error correcting performance enables efficient and reliable communication. However, a LDPC code decoding algorithm needs to be executed efficiently to meet cost, time, power and bandwidth requirements of target applications. The constructed codes should also meet error rate performance requirements of those applications. Since their rediscovery, there has been much research work on LDPC code construction and implementation. LDPC codes can be designed over a wide space with parameters such as girth, rate and length. There is no unique method of constructing LDPC codes. Existing construction methods are limited in some way in producing good error correcting performing and easily implementable codes for a given rate and length. There is a need to develop methods of constructing codes over a wide range of rates and lengths with good performance and ease of hardware implementability. LDPC code hardware design and implementation depend on the structure of target LDPC code and is also as varied as LDPC matrix designs and constructions. There are several factors to be considered including decoding algorithm computations,processing nodes interconnection network, number of processing nodes, amount of memory, number of quantization bits and decoding delay. All of these issues can be handled in several different ways. \nThis thesis is about construction of LDPC codes and their hardware implementation. LDPC code construction and implementation issues mentioned above are too many to be addressed in one thesis. The main contribution of this thesis is the development of LDPC code construction methods for some classes of structured LDPC codes and techniques for reducing decoding time. We introduce two main methods for constructing structured codes. In the first method, column-weight two LDPC codes are derived from distance graphs. A wide range of girths, rates and lengths are obtained compared to existing methods. The performance and implementation complexity of obtained codes depends on the structure of their corresponding distance graphs. In the second method, a search algorithm based on bit-filing and progressive-edge growth algorithms is introduced for constructing quasi-cyclic LDPC codes. The algorithm can be used to form a distance or Tanner graph of a code. This method could also obtain codes over a wide range of parameters. Cycles of length four are avoided by observing the row-column constraint. \nRow-column connections observing this condition are searched sequentially or randomly. Although the girth conditions are not sufficient beyond six, larger girths codes were easily obtained especially at low rates. The advantage of this algorithm compared to other methods is its flexibility. It could be used to construct codes for a given rate and length with girths of at least six for any sub-matrix configuration or rearrangement. The code size is also easily varied by increasing or decreasing sub-matrix size. Codes obtained using a sequential search criteria show poor performance at low girths (6 and 8) while random searches result in good performing codes. \nQuasi-cyclic codes could be implemented in a variety of decoder architectures. One of the many options is the choice of processing nodes interconnect. We show how quasi-cyclic codes processing could be scheduled through a multistage network. Although these net-works have more delay than other modes of communication, they offer more flexibility at a reasonable cost. Banyan and Benes networks are suggested as the most suitable networks. \nDecoding delay is also one of several issues considered in decoder design and implementation. In this thesis, we overlap check and variable node computations to reduce decoding time. Three techniques are discussed, two of which are introduced in this thesis. The techniques are code matrix permutation, matrix space restriction and sub-matrix row-column scheduling. Matrix permutation rearranges the parity-check matrix such that rows and columns that do not have connections in common are separated. This techniques can be applied to any matrix. Its effectiveness largely depends on the structure of the code. We show that its success also depends on the size of row and column weights. Matrix space restriction is another technique that can be applied to any code and has fixed reduction in time or amount of overlap. Its success depends on the amount of restriction and may be traded with performance loss. The third technique already suggested in literature relies on the internal cyclic structure of sub-matrices to achieve overlapping. The technique is limited to LDPC code matrices in which the number of sub-matrices is equal to row and column weights. We show that it can be applied to other codes with a lager number of sub-matrices than code weights. However, in this case maximum overlap is not guaranteed. We calculate the lower bound on the amount of overlapping. Overlapping could be applied to any sub-matrix configuration of quasi-cyclic codes by arbitrarily choosing \nthe starting rows for processing. Overlapping decoding time depends on inter-iteration waiting times. We show that there are upper bounds on waiting times which depend on the code weights. Waiting times could be further reduced by restricting shifts in identity sub-matrices or using smaller sub-matrices. This overlapping technique can reduce the decoding time by up to 50% compared to conventional message and computation scheduling. \nTechniques of matrix permutation and space restriction results in decoder architectures that are flexible in LDPC code design in terms of code weights and size. This is due to the fact that with these techniques, rows and columns are processed in sequential order to achieve overlapping. However, in the existing technique, all sub-matrices have to be processed in parallel to achieve overlapping. Parallel processing of all code sub-matrices requires the architecture to have the number of processing units at least equal to the number sub-matrices. Processing units and memory space should therefore be distributed among the sub-matrices according to the sub-matrices arrangement. This leads to high complexity or inflexibility in the decoder architecture. We propose a simple, programmable and high throughput decoder architecture based on matrix permutation and space restriction techniques.