Linear and non-linear codes that very closely approach the capacities of finite-state machine channels are presented. The independent and uniformly distributed (i.u.d.) information rates (capacities) of partial response channels are considered first. These information rates, obtained when the channel input is i.u.d., are achievable by linear coset codes. This is demonstrated by optimization of practical iteratively decodable low-density parity-check (LDPC) codes. The presented results show that the optimized LDPC codes approach the i.u.d. capacities of binary-input intersymbol interference channels to within 0.1--0.2dBs.; For many channels of interest the independent and uniformly distributed input is not optimal and, therefore, the i.u.d. capacity does not equal the capacity. As the first step in constructing codes that approach the capacities, a numerical method that computes the Markov capacities of the finite-state machine channels is proposed. The Markov capacities are computed for practical channels with finite-input alphabets and two input power constraints: the average power constraint and the peak-to-average power ratio constraint.; A concatenated coding scheme is formed to achieve the Markov capacities. The method is general and applies to any finite-state machine channel. The proposed concatenated code consists of an inner (non-linear) trellis code and an outer LDPC (coset) code. The inner code is constructed to mimic the optimized Markov source and achieves the Markov information rate that closely approaches the capacity. Hence, the inner code is named matched information rate (MIR) code. The outer code is optimized on the superchannel consisting of the MIR trellis code and the channel.; The final part of the thesis concentrates on the topic of decoding of LDPC codes with practical block lengths. A novel iterative decoding algorithm is proposed to bridge the gap between the standard iterative sum-product decoder and the optimal (but prohibitively complex) maximum-likelihood decoder. The method relies on information correction procedures implemented on select symbol nodes of LDPC code graphs. It is demonstrated that the proposed information correction decoder achieves considerable gains compared to the standard sum-product decoder, especially for short LDPC codes. With a sufficiently large number of decoding iterations, the presented information correction approach almost achieves the maximum-likelihood decoding performance.
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