The paper describes six detection processes that are developments of the Viterbi-algorithm detector and are suitable for use in a synchronous serial data-transmission system operating at a transmission rate of up to 20,000 bit/s over a slowly-time-varying channel. The processes are first described and the results of computer simulation tests are then presented, comparing the tolerances of the detection processes to additive white Gaussian noise with the tolerances of both the Viterbi-algorithm detector and a conventional non-linear transversal equalizer. Several different time-invariant channels are used in the tests. It is shown that, under the appropriate conditions, a near optimum tolerance to noise can be achieved by a detection process that involves very much less storage than the Viterbi-algorithm detector and requires only a very small fraction of the number of operations.