The paper describes a theoretical study of two different 19200 bit/s q.a.m. data-transmission systems suitable for use over voice-band telephone circuits. One of the systems employs a 64-point q.a.m. signal with an element rate of 3200 bauds, and the other employs a 256-point q.a.m. signal with an element rate of 2400 bauds. A near-maximum-likelihood detection process, derived from a reduced-state Viterbi-algorithm detector, is used at the receiver in each case. Results of computer-simulation tests are presented, comparing the tolerances of the two systems to additive white Gaussian noise, where the data are transmitted over models of four different telephone circuits. The effects, on the tolerance to noise, of inaccuracies in the estimates of the level and carrier phase of the received signal are also studied.