On optimal detectors in multiuser detection problems (Q2388467)

This paper is a supplement to [Probl. Inf. Transm. 39, No. 2, 191--206 (2003); translation from Prob. Peredachi Inf. 39, No. 2, 36--52 (2003; Zbl 1077.94006)]. Necessary and sufficient conditions under which asymptotically optimal detectors are linear are given. It is also shown that if, in contrast to [Zbl 1077.94006], optimal detectors are linear (moreover, nonasymptotically) when minimax, instead of Bayesian, statement of the problem with unknown coefficients is considered. A geometrical meaning of Theorem 1 from [Zbl 1077.94006] is explained, and it is shown that the theorem follows from some general results in [Teor. Veroyatn. Primen. 24, 106--118 (1979; Zbl 0418.60044)] and [\textit{M. V. Burnashev}, Math. Notes 32, 357--761 (1983; Zbl 0534.60006)] on hypotheses testing. It is also shown that some results of \textit{R. Lupas} and \textit{S. Verdú} [IEEE Trans. Inf. Theory 35, No. 1, 123--136 (1989; Zbl 0671.94003)] follow from [Zbl 1077.94006, Theorem 1].