In this paper, two 4$\times$4 rectangular patch array antennas operating at 20 GHz are implemented for the satellite communication. The sixteen patch antennas and microstrip feeding line are printed on a single-layered substrate. The design goal is to achieve high directivity and gain by optimizing design parameters through permutations in element spacing. The spacing between the array elements is chosen to be 0.736$\lambda$. Numerical simulation results indicate that the HPBW(Half-Power Beam Width) of the 4$\times$4 patch array antenna is 18.78 degrees in the E-plane and 18.48 degrees in the H-plane with a gain of 17.18 dBi. Numerical simulations of a 4$\times$4 recessed patch array antenna yield a HPBW of 18.71 degrees in the E-plane and 17.82 degrees in the H-plane with a gain of 19.43 dBi.