The use of high-temperature superconducting films for the fabrication of antennas allows us to realize radiating structures that are small in size and have a very high efficiency. In this paper, we investigate the properties of a novel compact antenna obtained by crossing a square patch with diagonal and transverse slots. Single patch antennas, with both linear and circular polarization operating in the X-band, with a size reduction of about 40% as compared to conventional square patches, have been designed and tested at prototype level. They have been realized by using double-sided YBa2Cu3O7-x,(YBCO) and Tl2Ba2Ca1Cu2O8(Tl-2212) superconducting films grown on MgO substrates and have been measured using a portable cryocooler. At T = 77 K, the experimental scattering parameter, S11, agrees well with simulations, showing a power handling of 23 dBm for YBCO devices. The miniaturized patch antenna investigated has allowed us to realize a 4 x 4 array of linear polarization antennas at 11.5 GHz, using a YBCO film grown on a MgO substrate with a 2" diameter, by considering a reduced interspace (more than 20%) between the elements. The far-field pattern and the 16-array gain are in good agreement with the simulated results.