In this paper, we use the class of non-binary hybrid LDPC codes to design very efficient low rate codes. To this end, we consider both asymptotic and finite length designs. First, we present an asymptotic analysis to design hybrid LDPC codes distributions and we explicit the cases when it is useful to use it. Then, by optimizing the algebraic properties of some topologies related to the Tanner graph, we design finite length codes. These codes show significant performance improvements both in the waterfall and in the error floor regions, in comparison with existing state-of-the-art low rate coding schemes, like turbo Hadamard or parallel concatenated Zigzag Hadamard codes, and with no increase of complexity. In particular, the minimum distance of low rate hybrid LDPC codes is by far greater than the ones of previously mentioned codes.
In this paper, we use the class of non-binary hybrid LDPC codes to design very efficient low rate codes. To this end, we consider both asymptotic and finite length designs. First, we present an asymptotic analysis to design hybrid LDPC codes distributions and we explicit the cases when it is useful to use it. Then, by optimizing the algebraic properties of some topologies related to the Tanner graph, we design finite length codes. These codes show significant performance improvements both in the waterfall and in the error floor regions, in comparison with existing state-of-the-art low rate coding schemes, like turbo Hadamard or parallel concatenated Zigzag Hadamard codes, and with no increase of complexity. In particular, the minimum distance of low rate hybrid LDPC codes is by far greater than the ones of previously mentioned codes.
Low-rate non-binary low-density parity-check (LDPC) codes for coherent and blockwise non-coherent additive white Gaussian noise (AWGN) channels are developed. The proposed construction is based on the concatenation of non-binary outer LDPC codes with inner binary codes. In case the binary codes are chosen to be Hadamard or Reed-Muller (RM) codes, the complexity of the decoding scheme is considerably reduced. An asymptotic analysis of the concatenation with help of composite capacity considerations anddensity evolution (DE) is provided, from which guidelines on the choice of both inner and outer codes are devised. Finite length designs presented in this work confirm the excellent performance of the proposed codes.
In this paper, some results on a very general class of LDPC codes are gathered. The hybrid LDPC codeword symbols belong to different order finite sets. That is why we can jointly control the couple (edge density,symbol size). Thanks to the stability condition given in this paper, we identify an advantage of hybrid LDPC codes over classical binary and non-binary LDPC codes. Information content evolution is analysed using a Gaussian approximation, in order to efficiently design hybrid LDPC codes. Some finite length constraints can be included in the asymptotic optimization, leading to good trade-off between convergence and error floor. In particular, for code rate 1/6, the optimized hybrid LDPC code is better than the best known codes.
In this paper, a new class of low-density parity-check (LDPC) codes, named hybrid LDPC codes, is introduced. Hybrid LDPC codes are characterized by an irregular connectivity profile and heterogeneous orders of the symbols in the codeword. It is shown in particular that the class of hybrid LDPC codes can be asymptotically characterized and optimized using density evolution (DE) framework, and a technique to maximize the minimum distance of the code is presented. Numerical assessment of hybrid LDPC code performances is provided, by comparing them to protograph-based and multiedge-type (MET) LDPC codes. Hybrid LDPC codes are shown to allow to achieve an interesting tradeoff between good error-floor performance and good waterfall region with nonbinary coding techniques.
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.
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.
This paper continues a previous work on non-binary cluster-LDPC codes. Such codes are defined by locally dense parity-check matrices, with (possibly rectangular) dense clusters of bits, but which are cluster-wise sparse. We derive a lower bound on the minimum distance of non-binary cluster-LDPC codes that is based on the topological properties of the underlying bipartite graph. We also propose an optimization procedure, which allows designing finite length codes with large minimum distance, even in the extreme case of codes defined by ultra-sparse graphs, i.e. graphs with all symbol-nodes of degree dv= 2. Furthermore, we provide asymptotic thresholds of ensembles of non-binary cluster-LDPC codes, which are computed exactly under the Belief Propagation decoding, and upper-bounded under the Maximum a Posteriori (MAP) decoding. We show that the MAP-threshold upper bounds, which are conjunctured to be tight, quickly approach the channel capacity, which confirms the excellent minimal distance properties of non-binary cluster-LDPC codes.