On optimum M-phase spreading sequences of Markov chains

We discuss designing M-phase spreading sequences of Markov chains optimal in terms of bit error probabilities, where M = 3, 4,... We consider SSMA (spread spectrum multiple access) communication systems using M-phase spreading sequences of Markov chains and give a necessary and sufficient condition that the limit of variance of the real part of the normalized MAI (multiple-access interference) for such systems becomes independent of the phase shifts. We also give a necessary and sufficient condition that the distribution of the real part of the normalized MAI is Gaussian. Based on these results, we discuss the optimum 3-phase and Q-phase (quadratic-phase) spreading sequences of Markov chains in terms of bit error probabilities in SSMA communication systems.