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Simulation and measurement results are presented for two designs of high-gain linearly polarized microstrip patch antennas, operating at 24 GHz on two high-permittivity substrates: Taconic CER-10 and high-resistivity silicon. One of these designs is new. In both designs, a multilayer construction is used and the patch is coupled to a coplanar waveguide (CPW) feed through a slot. The problem of surface waves is avoided by introducing an air gap between the slot and the patch with the help of micromachining or spacer layer; in one of the designs, the superstrate effect is also employed. For the new design, the measured gain value in the forward direction exceeds 6 dBi.