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.
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.
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.
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.
In 1948, Claude Shannon provided a sound mathematical foundation for communication, and proved the existence of good codes. Low-density parity-check (LDPC) codes are a family of error-correcting codes that achieve reliable performance (i.e., arbitrarily low probability of decoding error) near the Shannon-limit when decoded iteratively via very efficient graph-based message-passing decoders, the complexity of which grows only linearly in the block-length. Despite their tremendous success, LDPC codes lack a strong theoretical foundation and few explicit constructions of them are known that outperform their random counterparts. The first part of this thesis investigates how the structure of a graph representing an LDPC code affects the decoding performance. In particular, lower bounds on the minimum pseudocodeword weight, a key parameter in predicting iterative decoding performance, are obtained which link the performance of the code to the structural properties of its graph. One bound is a tree bound and is extended to generalized and p-ary LDPC codes. To this end, a suitable definition of pseudocodeword weight is derived for the p-ary symmetric channel. Moreover, the relationship between pseudocodeword weight distributions and different graph representations is examined. Bounds on the minimum pseudocodeword weight are also presented for several classes of expander codes based on the eigenvalues of the adjacency matrix of the Tanner graph. The second part of this thesis constructs LDPC codes having desirable properties. The first construction is combinatorial and uses permutations and mutually orthogonal Latin squares. One type results in the well-known projective-geometrybased LDPC codes. Another type yields a new family of codes having comparable parameters and pseudocodeword weights suitable for iterative decoding. The resulting performance of these codes is superior in comparison with that of comparable random LDPC codes, a feature attributed to the lack of low-weight bad pseudocodewords. For the second construction, unbalanced expander graphs are obtained by modifying the original zig-zag graph product to take bi-regular expander graphs as components, with suitable constraints on the degrees. Graphs resulting from the zig-zag and replacement product graphs are then used in the design of generalized LDPC codes having compact representation and pseudocodeword weights lower-bounded by the eigenvalue bounds.
This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum distances than equivalent length and rate randomly constructed LDPC codes. Further, by exploiting the resolvability of oval designs, and also by employing column splitting, we are able to produce 4-cycle free LDPC codes for a wide range of code rates and lengths while maintaining code regularity. Copyright # 2003 AEI. 1.
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.
Low-density parity-check (LDPC) block codes invented by Gallager in 1962 achieve exceptional error performance on a wide variety of communication channels. LDPC convolutional codes are the convolutional counterparts of LDPC block codes. This dissertation describes techniques for the design of LDPC convolutional codes, and analyzes their distance properties and iterative decoding convergence thresholds. The construction of time invariant LDPC convolutional codes by unwrapping the Tanner graph of algebraically constructed quasi-cyclic LDPC codes is described. The convolutional codes outperform the quasi-cyclic codes from which they are derived. The design of parity-check matrices for time invariant LDPC convolutional codes by the polynomial extension of a base matrix is proposed. An upper bound on free distance, proving that time invariant LDPC codes are not asymptotically good, is obtained. The Tanner graph is used to describe a pipelined message passing based iterative decoder for LDPC convolutional codes that outputs decoding results continuously. The performance of LDPC block and convolutional codes are compared for fixed decoding parameters like computational complexity, processor complexity, and decoding delay. In each case, the LDPC convolutional code performance is at least as good as that of LDPC block codes. An analog circuit to implement pipelined decoding of LDPC convolutional codes is proposed. The distance properties of a permutation matrix based (time varying) ensemble of (J,K) regular LDPC convolutional codes is investigated. It is proved that these codes (for J > 2) have free distance increasing linearly with constraint length, i.e., they are asymptotically good. Further, the asymptotic free distance to constraint length ratio for the convolutional codes is several times larger than the minimum distance to block length ratio for corresponding LDPC block codes. Iterative decoding of terminated LDPC convolutional codes, based on the ensemble mentioned above, is analyzed, assuming transmission over a binary erasure channel. The structured irregularity of the codes leads to significantly better convergence thresholds compared to corresponding LDPC block codes. At the calculated thresholds, both the bit and block error probability can be made arbitrarily small. The results obtained suggest that the thresholds approach capacity with increasing J.