We apply the density-evolution technique to determine the thresholds of low-density parity-check (LDPC) codes when the sum-product algorithm is employed to perform joint channel-state estimation and decoding. The channel considered is the two-state noiseless/useless binary symmetric channel (BSC) block interference channel, where a block of h consecutive symbols shares the same channel state, which is either a noiseless BSC (crossover probability 0) or a useless BSC (crossover probability 1/2). The channel state is selected independently and at random from block to block, according to a known prior distribution. The threshold of the joint channel-state estimation/decoding scheme when used over such a channel is shown to be greatly superior to that of a decoder that makes no attempt to estimate the channel state. These results are also confirmed by simulation. The maximum-likelihood (ML) performance of LDPC codes when used over this channel is investigated. Lower bounds on the error exponents of regular LDPC codes, when ML decoded, are shown to be close to the random coding channel error exponent when the LDPC variable node degree is high.

We apply low-density parity-check (LDPC) codes to a bandwidth-efficient modulation scheme using multilevel coding, multistage decoding, and trellis-based signal shaping. Performance results based on density evolution and simulations are presented. Using irregular LDPC component codes of block length 10/sup 5/ and a 64-quadrature amplitude modulation signal constellation operating at 2 bits/dimension, a bit-error rate of 10/sup -5/ is achieved at an E/sub b//N/sub 0/ of 6.55 dB. At this value of E/sub b//N/sub 0/, the Shannon channel capacity, computed assuming equally likely signaling, is below 2 bits/dimension.

A very noisy channel (VNC), is a discrete-input memoryless channel whose capacity is close to zero. Very noisy channels are of interest, since they serve as models for some important physical channels. There are two distinct classes of VNCs: Reiffen's class I (Reiffen, 1963) and Majani's class II (Majani, 1987). It is shown that the reliability function is known exactly for both classes of VNCs, by extending the results previously obtained only for class I. It is then shown that an exponentially optimum code can be constructed for a channel if it can be modeled as repeated uses of a binary-input class I VNC, a binary-input/binary-output class II VNC or a class II very noisy binary erasure channel. The theory developed is illustrated by considering the direct detection optical channel used with polarization modulation. The capacity, reliability function, and exponentially optimum code, for this channel, are derived.>