This paper presents a study on generalized synchronization processes in one-dimensional cellular automata. The authors investigate the problem of synchronizing large-scale cellular automata with an initial general located at any generalized position. They focus on fundamental generalized firing squad synchronization algorithms operating in optimum- and non-optimum-steps on one-dimensional cellular arrays. A new eight-state algorithm is proposed, which is the smallest known in the class of generalized optimum-step firing squad synchronization protocols. The firing squad synchronization problem (FSSP) is a classic problem in cellular automata, where the goal is to synchronize all cells in a one-dimensional array with a general cell at one end. The problem has been studied extensively for over 40 years, with various algorithms proposed to solve it. The authors examine several existing algorithms, including the 17-state algorithm by Moore and Langdon, the ten-state algorithm by the three Russians, and the nine-state algorithms by Szwerinski, Settle and Simon, and UHMNM. They also present a new eight-state algorithm and a six-state non-optimum-step algorithm. The authors analyze the state transition rules of these algorithms, focusing on the number of states, transition rules, and the filled-in ratio. They find that the eight-state algorithm has the smallest number of states among the generalized optimum-step algorithms. They also examine the state-change complexity of these algorithms, finding that they all have an O(n²) complexity. The paper also discusses the theoretical aspects of the FSSP, including lower bounds on the minimum time required for synchronization, the number of states required for a full solution, and the computational complexity of synchronization algorithms. The authors conclude that the eight-state algorithm is the most efficient among the generalized optimum-step algorithms, and that further research is needed to find even more efficient solutions.This paper presents a study on generalized synchronization processes in one-dimensional cellular automata. The authors investigate the problem of synchronizing large-scale cellular automata with an initial general located at any generalized position. They focus on fundamental generalized firing squad synchronization algorithms operating in optimum- and non-optimum-steps on one-dimensional cellular arrays. A new eight-state algorithm is proposed, which is the smallest known in the class of generalized optimum-step firing squad synchronization protocols. The firing squad synchronization problem (FSSP) is a classic problem in cellular automata, where the goal is to synchronize all cells in a one-dimensional array with a general cell at one end. The problem has been studied extensively for over 40 years, with various algorithms proposed to solve it. The authors examine several existing algorithms, including the 17-state algorithm by Moore and Langdon, the ten-state algorithm by the three Russians, and the nine-state algorithms by Szwerinski, Settle and Simon, and UHMNM. They also present a new eight-state algorithm and a six-state non-optimum-step algorithm. The authors analyze the state transition rules of these algorithms, focusing on the number of states, transition rules, and the filled-in ratio. They find that the eight-state algorithm has the smallest number of states among the generalized optimum-step algorithms. They also examine the state-change complexity of these algorithms, finding that they all have an O(n²) complexity. The paper also discusses the theoretical aspects of the FSSP, including lower bounds on the minimum time required for synchronization, the number of states required for a full solution, and the computational complexity of synchronization algorithms. The authors conclude that the eight-state algorithm is the most efficient among the generalized optimum-step algorithms, and that further research is needed to find even more efficient solutions.