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We present an analysis under the iterative decoding of coset low-density parity-check (LDPC) codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random- coset analysis to produce an effect that is similar to output symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q-1-dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop extrinsic information transfer (EXIT) charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the additive white Gaussian noise (AWGN) channel, including results within 0.56 dB of the unrestricted Shannon limit (i.e., not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz.