A satellite communication terminal, comprising: an antenna system comprising a satellite antenna (213; 213A) including a plurality of antenna elements (213B; 301), each antenna element comprising: one or more first terminals that receive a vertically polarized wireless signal; one or more second terminals that receive a horizontally polarized wireless signal; and a processing circuit that combines the vertically polarized signal and the horizontally polarized signal using a combiner after applying a 90-degree phase shift to the vertically polarized signal or the horizontally polarized signal to obtain a left-hand circularly polarized (LHCP) signal or a right-hand circularly polarized (RHCP) signal; a processor that controls the components of the satellite communication terminal; and a memory, associated with the processor, that stores a lookup table formed by results of a plurality of off-line computations, wherein the plurality of antenna elements are asymmetrically positioned and asymmetrically oriented in a flat antenna array to form a geometrical structure, wherein the geometrical structure of the antenna array is characterized in terms of multiple parameters, the multiple parameters of the geometrical structure being capable of being optimized off-line in conjunction with a plurality of quantized phase shift values to shape a plurality of antenna patterns that include a best antenna pattern, wherein the lookup table includes the plurality of antenna patterns that are based on a parametric model that relies on the geometrical structure of the flat antenna array, and wherein the satellite communication terminal is configured to steer a beam of the flat antenna array to the best antenna pattern by: characterizing the geometrical structure of the flat antenna array in terms of multiple parameters; optimizing the multiple parameters of the geometrical structure off-line in conjunction with the plurality of quantized phase shift values to shape the plurality of antenna patterns that include the best antenna pattern; indexing the lookup table by an input vector comprising multiple variables to determine a selection of the best antenna pattern; and steering the beam of the antenna array to the best antenna pattern. an antenna system comprising a satellite antenna (213; 213A) including a plurality of antenna elements (213B; 301), each antenna element comprising: one or more first terminals that receive a vertically polarized wireless signal; one or more second terminals that receive a horizontally polarized wireless signal; and one or more first terminals that receive a vertically polarized wireless signal; one or more second terminals that receive a horizontally polarized wireless signal; and a processing circuit that combines the vertically polarized signal and the horizontally polarized signal using a combiner after applying a 90-degree phase shift to the vertically polarized signal or the horizontally polarized signal to obtain a left-hand circularly polarized (LHCP) signal or a right-hand circularly polarized (RHCP) signal; a processor that controls the components of the satellite communication terminal; and a memory, associated with the processor, that stores a lookup table formed by results of a plurality of off-line computations, wherein the plurality of antenna elements are asymmetrically positioned and asymmetrically oriented in a flat antenna array to form a geometrical structure, wherein the geometrical structure of the antenna array is characterized in terms of multiple parameters, the multiple parameters of the geometrical structure being capable of being optimized off-line in conjunction with a plurality of quantized phase shift values to shape a plurality of antenna patterns that include a best antenna pattern, wherein the lookup table includes the plurality of antenna patterns that are based on a parametric model that relies on the geometrical structure of the flat antenna array, and wherein the satellite communication terminal is configured to steer a beam of the flat antenna array to the best antenna pattern by: characterizing the geometrical structure of the flat antenna array in terms of multiple parameters; optimizing the multiple parameters of the geometrical structure off-line in conjunction with the plurality of quantized phase shift values to shape the plurality of antenna patterns that include the best antenna pattern; indexing the lookup table by an input vector comprising multiple variables to determine a selection of the best antenna pattern; and steering the beam of the antenna array to the best antenna pattern.