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Summary form only given. A "sphere decoding" algorithm is suggested for finding the ML solution with a specified (high) probability; it is closely related to the Wozencraft sequential decoding algorithm. Closed-form expressions for the expected complexity are then found by using and extending some classical number-theoretic results and techniques. A multiple (transmit and receive) antenna problem is studied in some detail. It is shown that for a wide range of signal-to-noise ratios (SNRs)and number of antennas, the expected complexity is polynomial, in fact often roughly cubic. This is the complexity of heuristic detection schemes such as the BLAST scheme developed at Bell Laboratories by G. Foschini and his colleagues. However, now for the same complexity we have an optimum solution, which yields much better error probability performance, thus allowing much lower SNR or many fewer antennas than with the heuristic schemes. The results can be extended to frequency selective channels, as well as to other communication problems. It is shown that expected complexity provides a new measure for evaluating and comparing different kinds of codes, such as turbo codes vs. LDPC codes.