Genetic algorithms have several important features that predestinate them to solving synthesis problems. The main aim of this paper is to show how to apply the GAs for solving the synthesis problems. We propose a number of concepts for enhancement of the GA?s effectiveness and efficiency. These concepts include the mixed selection mechanisms, deterministic crossover and mutation operators, pseudo-random construction of the initial population, evolution of the application probabilities of operators with the progress of computations etc. In the paper, an effective and efficient GA scheme is proposed and applied for solving an important design problem: the minimal input support problem (MISP). Our GA produces in almost all cases the strictly optimal results and realizes the best trade-off between the effectiveness and efficiency. The experimental results clearly demonstrate that the proposed GA scheme is suitable for solving the synthesis problems and its application results in very effective and efficient genetic synthesis algorithms.