Research on LDPC codes is in the ascendant nowadays.Our paper has done some work on the proposition ofperformance of different codes in the same family constrained by weight distribution polynomials.Different codes of the same weight distribution are listed,and the results indicate the upper performance bound is determined by density evolution theory and the lower performance bound by the Fill-Shift construction method.Furthermore,the codes in the set of elementary matrix transforma- tion share the same error-correction performance but different coding complexity.We can get a conclusion that codes constrained by weight distribution polynomial can be divided into many groups,performance among different groups is not the same,it depends on the loops length and distribution,and performance in the same group is identical.What we can do now is that to find the best group and do suitable transformations if given weight distribution polynomials.Through this paper,we can see that a new idea for code constructing is also presented.If the relations between big girth of LDPC codes and weight distribution polynomial are discovered, new codes can be achieved only by the weight distribution polynomial,the currently used method given in references for code con- structing will be less useful.We can think that we have found a completely different method to construct LDPC codes.
Research on LDPC codes is in the ascendant nowadays.Our paper has done some work on the proposition ofperformance of different codes in the same family constrained by weight distribution polynomials.Different codes of the same weight distribution are listed,and the results indicate the upper performance bound is determined by density evolution theory and the lower performance bound by the Fill-Shift construction method.Furthermore,the codes in the set of elementary matrix transforma- tion share the same error-correction performance but different coding complexity.We can get a conclusion that codes constrained by weight distribution polynomial can be divided into many groups,performance among different groups is not the same,it depends on the loops length and distribution,and performance in the same group is identical.What we can do now is that to find the best group and do suitable transformations if given weight distribution polynomials.Through this paper,we can see that a new idea for code constructing is also presented.If the relations between big girth of LDPC codes and weight distribution polynomial are discovered, new codes can be achieved only by the weight distribution polynomial,the currently used method given in references for code con- structing will be less useful.We can think that we have found a completely different method to construct LDPC codes.
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.
There is an important breakthrough in the field of forward error correction techniques, low-density parity-check (LDPC) codes. These codes have potential to approach the Shannon limit with reliable performance on a given channel, known as the channel capacity. Recently, researchers pay much more attention towards LDPC codes because its performance is better than Turbo codes. However, the LDPC code is the best when it is an irregular code and uses with a very large block-length. Although, the short block-length LDPC codes perform not so well, they are easy to implement in various practical applications. Therefore, there is still a demand of new development on the encoding side of LDPC codes for various ranges of code rate and code length so as to capable of using in many applications.This thesis proposes the development of quasi-cyclic (QC) LDPC codes for column weight 3. In the first part of this thesis, we propose a new construction algorithm of QC-LDPC codes for medium to large block-length by combining QC-LDPC codes of small block-length as their component codes, via Chinese remainder theorem (CRT). Such component codes were constructed by permuting each column block sequentially to attain the desire local girth. After combining all component codes to generate an expanded parity-check matrix, the resulting girth is greater than or at least equal to the maximum girth of the component codes. Simulation results show that our proposed construction method of the parity-check matrix significantly outperforms the other well-known existing methods in terms of low error-floor, simple structure, high performance, and can reduce encoding complexity.In addition, this thesis also proposes two new construction methods for QC-LDPC codes, namely a base matrix method and a subtraction based method. A base matrix based method is a simple, less computational complexity method for constructing the exponent matrix (3, K) of girth 8, 10 and 12 of QC-LDPC codes. Another method is a subtraction method, which has a similar exponent matrix as a base matrix method and has a girth at least 8. Results indicate that the LDPC codes constructed from these two methods have flexibility for arbitrary block-column length K and have a similar BER performance if compared to existing methods.
Low-density parity-check (LDPC) codes achieve outstanding performance by the sum-product decoder. Its performance approaches the maximum-likelihood (ML) decoding performance at high SNR region. Upper bounds on the error probability of the ML decoded LDPC codes can be computed using the weight distribution. Recently, X.-Y. Hu et al. presented an approximate algorithm for estimating the weight distribution of LDPC codes. However, the weight distribution computed by their method depends on the iterative decoding algorithm, and is not always estimated exactly. In this paper, we propose a probabilistic method for computing the low-part weight distribution of LDPC codes. Using our method, the approximate weight distribution of LDPC codes can be computed with high accuracy
Since the weight and distance computations of LDPC codes is a NP problem, most presented evaluations of LDPC codes are based on the girth numbers of the codes, and the computation for the distance of the codes is also depended on the girth numbers. There has been no simple way to obtain the distance of LDPC codes because of the code length. Based on generation matrices, this paper develops an algorithm to estimate the upper bound of minimum weights and minimum distance of LDPC codes. The paper establishes the upper bounds of minimum weights and distances of LDPC codes by using the vectors of generation matrices, which is different from the presented probabilitistic search methods for the weights' and distances of LCPC codes. The proposed algorithm can greatly reduce the searching time of weights and distances. Applying the algorithm we find that some LDPC codes get good BER performance although they are only free girth 4 but of great distances. Simulations verify the algorithms to be valid.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are one of the most important code classes of LDPC codes. They have two drawbacks: lack of randomness and limited girth lead to a degraded decoding performance in the waterfall and error floor regions, respectively. To tackle these problems, we present a new class of LDPC codes, named linear permutation polynomial (LPP) codes, whose parity-check matrix consists of permutation matrices based on LPPs. The girth of regular QC-LDPC codes is upper bounded by 12, while LPP codes break this limit. We demonstrate that LPP codes have error performance almost equivalent to random ones.
Low-density parity-check (LDPC) codes have higher coding gains compared to algebraic codes in several configurations. However, unlike algebraic codes whose performance can be easily determined generally, the block error rate performance of an LDPC code needs to be obtained by simulations. While there are analytical methods for some structured LDPC codes that predict this performance in the low error rate regime, these provide little insight into the error floor of other (less structured) LDPC codes. In this article, we present a simple method to upper bound the error floor of LDPC codes based on a certain monotonicity assumption on the decoder. This upper bound is general enough so that under the monotonicity assumption it is valid for any error-correcting code. This upper bounding technique is useful when projecting very low block error rates (such as below 1E-14) for LDPC codes and there is no knowledge of the trapping sets.