In this correspondence, we investigate in a comprehensivefashion a one-layer coding/shaping scheme resembling a perfectly cooperatedmultiple-access system. At the transmitter, binary data areencoded by either single-level or multilevel codes. The coded bits are firstrandomly interleaved and then entered into a signal mapper. At each time,the signal mapper accepts as input multiple binary digits and deliversas output an amplitude signal, where the input are first independentlymapped into 2-PAM signals (possibly having different amplitudes) andthen superimposed to form the output. The receiver consists of an iterativedecoding/demapping algorithm with an entropy-based stopping criterion.In the special cases when all the 2-PAM signals have equal amplitudes,based on an irregular trellis, we propose an optimal soft-input–soft-output(SISO) demapping algorithm with quadratic rather than exponentialcomplexity. In the general cases, when multilevel codes are employed,we propose power-allocation strategies to facilitate the iterative decoding/dempaping algorithm. Using the unequal power-allocations andthe Gaussian-approximation-based suboptimal demapping algorithm(with linear complexity), coded modulation with high bandwidth efficiencycan be implemented.

We study the limits of performance of Gallager codes(low-density parity-check (LDPC) codes) over binary linear intersymbolinterference (ISI) channels with additive white Gaussiannoise (AWGN). Using the graph representations of the channel,the code, and the sum-product message-passing detector/decoder,we prove two error concentration theorems. Our proofs expandon previous work by handling complications introduced by thechannel memory. We circumvent these problems by consideringnot just linear Gallager codes but also their cosets and by distinguishingbetween different types of message flow neighborhoodsdepending on the actual transmitted symbols. We compute thenoise tolerance threshold using a suitably developed densityevolution algorithm and verify, by simulation, that the thresholdsrepresent accurate predictions of the performance of the iterativesum-product algorithm for finite (but large) block lengths. Wealso demonstrate that for high rates, the thresholds are very closeto the theoretical limit of performance for Gallager codes overISI channels. If C denotes the capacity of a binary ISI channeland if Ci i d denotes the maximal achievable mutual informationrate when the channel inputs are independent and identicallydistributed (i.i.d.) binary random variables (Ci i d C), weprove that the maximum information rate achievable by thesum-product decoder of a Gallager (coset) code is upper-boundedby Ci i d . The last topic investigated is the performance limit ofthe decoder if the trellis portion of the sum-product algorithm isexecuted only once; this demonstrates the potential for tradingoff the computational requirements and the performance of thedecoder.