A bstruci-A method is described for constructing long error-correcting codes from one or more shorter error-correcting codes, referred to as subcodes, and a bipartite graph. A graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes. Lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subcodes. Both the encoders and decoders proposed are shown to take advantage of the code's explicit decomposition into subcodes to decompose and simplify the associated computational processes. Bounds on the performance of two specific decoding algorithms are established, and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed. The proposed decoders are able to mahe effective use of probabilistic information supplied by the channel receiver, e.g., reliability information, without greatly increasing the number of computations required. It is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst-error correction abilities. The construction principles are illustrated by several examples.

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described. Index Terms-Circulant matrices, iterative decoding, low-density parity-check (LDPC) block codes, LDPC convolutional codes, LDPC codes, message-passing, quasi-cyclic (QC) codes. I. INTRODUCTION L OW-density parity-check (LDPC) codes have attracted considerable attention in the coding community because they can achieve near-capacity performance with iterative message-passing decoding and sufficiently long block sizes. For example, in [1], Chung et al. presented a block length (ten million bits) rate-LDPC code that achieves reliable performance-a bit error rate (BER)-on an additive white Gaussian noise (AWGN) channel with a signal-to-noise ratio (SNR) within 0.04 dB of the Shannon limit. For many practical applications, however, the design of good codes with shorter block lengths is desired. Moreover,

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The parity-check matrix of a linear code is used to define a bipartite code constraint (Tanner) graph in which bit nodes are connected to parity check nodes. The connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix. A simple lower bound on minimum distance of the code is expressed in terms of the two largest eigenvalues. For a more powerful bound, local properties of the subgraph corresponding to a minimum weight word in the code are used to create an optimization problem whose solution is a lower bound on the code's minimum distance. Linear programming gives one bound. The technique is illustrated by applying it to sparse block codes with parameters [7,3,4] and [42,23,6].

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Random walks with step number fluctuations are examined in n dimensions for when step lengths comprising the walk are governed by stable distributions, or by random variables having power-law tails. When the number of steps taken in the walk is large and uncorrelated, the conditions of the Lévy-Gnedenko generalization of the central limit theorem obtain. When the number of steps is correlated, infinitely divisible limiting distributions result that can have Lévy-like behavior in their tails but can exhibit a different power law at small scales. For the special case of individual steps in the walk being Gaussian distributed, the infinitely divisible class of K distributions result. The convergence to limiting distributions is investigated and shown to be ultraslow. Random walks formed from a finite number of steps modify the behavior and naturally produce an inner scale. The single class of distributions derived have as special cases, K distributions, stable distributions, distributions with power-law tails, and those characteristic of high and low frequency cascades. The results are compared with cellular automata simulations that are claimed to be paradigmatic of self-organized critical systems.