Coded modulation systems based on low density parity check (LDPC) codes of finite lengths are considered. The union bounds on the bit error probabilities of the maximum likelihood (ML) decoding are presented for both additive white Gaussian noise (AWGN) and flat Rayleigh fading channels. The tightness of the derived bound is verified by simulating the ML decoding of a very short LDPC code. For medium-length codes, performance of the sum-product decoding can asymptotically approach the bounds.

This letter considers multiple access systems without bandwidth expansion. To improve the spectral efficiency, each user employs a QPSK modulation. The orientation of QPSK constellations is designed to maximize the minimum distance of the superimposed symbol constellation. The upper and lower bounds for the error performance of the proposed design demonstrate its advantage.