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On a Class of High-Girth LDPC Codes Based on Finite Multidimensional Lattices

John Craddock, Mark F. Flanagan, Anthony Fagan

Published Jun 1, 2011• 1 citation
OpenAlex

Abstract

An LDPC code construction technique is proposed based on the structural properties of finite m-dimensional lattices. The Tanner graph of any code from this class is shown to have a girth of eight, and the number of proper eightcycles in the graph is enumerated. The minimum distance of the codes is shown to be lower bounded by 2 m. The codes are also shown to be highly flexible in terms of code length and rate, and compatible with a low-complexity serial-parallel decoder implementation based on the turbo-decoding message passing algorithm. Finally, simulation results over the AWGN channel demonstrate that these codes have good error-correcting performance. 1

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A Euclidean Geometry Based Algebraic Construction Technique for Girth-8 Gallager LDPC Codes

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A construction technique is proposed for low-density parity-check (LDPC) codes based on finite Euclidean geometries EG(m,2s). These codes are shown to be regular Gallager codes with Tanner graphs of girth eight. The minimum distance of these codes is shown to be lower-bounded by 2m. The codes are also amenable to an efficient partly parallel decoder implementation, which may be used in conjunction with the turbo decoding message passing (TDMP) algorithm for LDPC decoding. Finally, simulation results show that these codes have very good error-correcting performance

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This paper presents a class of regular quasi-cyclic (QC) LDPC codes whose Tanner graphs have girth at least eight. These codes are constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes with minimum distance 5. Masking their parity-check matrices significantly reduces the numbers of short cycles in their Tanner graphs and results in codes which perform well over the AWGN channel in both waterfall and low error-rate regions.

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Flexible Construction of High-Girth QC-LDPC Codes

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This article presents a highly flexible method for constructing high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes. The proposed algorithm constructs a Tanner graph formed by connecting groups of rows (check nodes) and columns (variable nodes) of the constructed code. To obtain a sub-matrix structure, rows and columns are divided into groups of equal sizes. Rows and columns in a group are connected in their numerical (positional) order to obtain a cyclic structure. Connected rows and columns must satisfy the desired minimum cycle length. We present conditions that guarantee desired girths. The proposed algorithm is by far more flexible in constructing a wide range (rates and lengths) of regular and irregular QC-LDPC codes compared to existing methods. The algorithm, which has linear complexity with respect to the number of rows or columns, provides an easy and fast way to construct QC-LDPC codes. Constructed codes show good bit error rate performances.

International journal of computer science and applications•Aug 1, 2012•2 citations

Low-Density Parity-Check Codes: Construction and Implementation

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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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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. 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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. 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Karunakaran Murali Krishnan, Rajdeep Singh, L. Sunil Chandran +1 more

An elementary combinatorial Tanner graph construction for a family of near-regular low density parity check (LDPC) codes achieving high girth is presented. These codes are near regular in the sense that the degree of a left/right vertex is allowed to differ by at most one from the average. The construction yields in quadratic time complexity an asymptotic code family with provable lower bounds on the rate and the girth for a given choice of block length and average degree. The construction gives flexibility in the choice of design parameters of the code like rate, girth and average degree. Performance simulations of iterative decoding algorithm for the AWGN channel on codes designed using the method demonstrate that these codes perform better than regular PEG codes and MacKay codes of similar length for all values of Signal to noise ratio.

arXiv:cs/0609146•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•5 citations

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Y. Kou, Shu Lin, M.P.C. Fossorier

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.

IEEE Transactions on Information Theory•Jan 1, 2001•1,303 citations
An LDPC code construction technique is proposed based on the structural properties of finite m-dimensional lattices. The Tanner graph of any code from this class is shown to have a girth of eight, and the number of proper eightcycles in the graph is enumerated. The minimum distance of the codes is shown to be lower bounded by 2 m. The codes are also shown to be highly flexible in terms of code length and rate, and compatible with a low-complexity serial-parallel decoder implementation based on the turbo-decoding message passing algorithm. Finally, simulation results over the AWGN channel demonstrate that these codes have good error-correcting performance. 1
A construction technique is proposed for low-density parity-check (LDPC) codes based on finite Euclidean geometries EG(m,2s). These codes are shown to be regular Gallager codes with Tanner graphs of girth eight. The minimum distance of these codes is shown to be lower-bounded by 2m. The codes are also amenable to an efficient partly parallel decoder implementation, which may be used in conjunction with the turbo decoding message passing (TDMP) algorithm for LDPC decoding. Finally, simulation results show that these codes have very good error-correcting performance
This article presents a highly flexible method for constructing high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes. The proposed algorithm constructs a Tanner graph formed by connecting groups of rows (check nodes) and columns (variable nodes) of the constructed code. To obtain a sub-matrix structure, rows and columns are divided into groups of equal sizes. Rows and columns in a group are connected in their numerical (positional) order to obtain a cyclic structure. Connected rows and columns must satisfy the desired minimum cycle length. We present conditions that guarantee desired girths. The proposed algorithm is by far more flexible in constructing a wide range (rates and lengths) of regular and irregular QC-LDPC codes compared to existing methods. The algorithm, which has linear complexity with respect to the number of rows or columns, provides an easy and fast way to construct QC-LDPC codes. Constructed codes show good bit error rate performances.
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.
An elementary combinatorial Tanner graph construction for a family of near-regular low density parity check (LDPC) codes achieving high girth is presented. These codes are near regular in the sense that the degree of a left/right vertex is allowed to differ by at most one from the average. The construction yields in quadratic time complexity an asymptotic code family with provable lower bounds on the rate and the girth for a given choice of block length and average degree. The construction gives flexibility in the choice of design parameters of the code like rate, girth and average degree. Performance simulations of iterative decoding algorithm for the AWGN channel on codes designed using the method demonstrate that these codes perform better than regular PEG codes and MacKay codes of similar length for all values of Signal to noise ratio.
This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.