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A class of optimum aperture distributions that gives the best mean-square approximation with weight factor ( to any given pattern space factor for an arbitrarily prescribed value of a generalized superdirectivity ratio is derived for all . For the parameter becomes Taylor's superdirectivity ratio and the -constrained solution reduces simply to a - constrained solution published earlier. But for , or 1 the solution becomes the best, or near best, -constrained solution in the sense of least radiated power in the error pattern for the case of -plane strip source antennas or of -plane strip or line-source antennas, respectively. Thus, for strip or line source antennas the new solution possesses an important physical interpretation that the earlier solution did not.