Complex binary sequences are generated through the application of simple threshold, linear transformations to the logistic iterative map. Depending primarily on the value of its non-linearity parameter, the logistic map exhibits a great variety of behavior, including stable states, cycling and periodical activity and the period doubling phenomenon that leads to high-order chaos. From the real data sequences, binary sequences are derived. Consecutive LL bit sequences are given as input to a cellular automaton with the task to regenerate the subsequent LL bits of the binary sequence in precisely LL evolution steps. To perform this task a genetic algorithm is employed to evolve cellular automaton rules. Various complex binary sequences are examined, for a variety of initial values and a wide range of values of the non-linearity parameter. The proposed hybrid multiple-step-ahead prediction algorithm, based on a combination of genetic algorithms and cellular automata proved efficient and effective.