One Dimensional Celular Automa Under Null Boundary Condition

In this study, we examine one-dimensional cellular automata under the null boundary condition. A matrix is created to represent the state of each cell using a local rule. This matrix contains coefficients that define the state transitions of each cell. The representative matrix depicts the system’s state as a vector and transformation rules are applied through matrix multiplications. This method allows us to analyze how the system evolves over time. When performing calculations over finite fields, we utilize matrix algebra. First, we define one-dimensional cellular automata. Then, we define our local rule. Using the local rule and matrix algebra, we obtain our representative matrix. While obtaining the representative matrix, we utilize natural bases. In previous studies, representative matrices were generally obtained by taking the radius as 1. Unlike other studies, we take the radius as 2. By taking the radius as 2, all the results we obtain will differ from those in other studies. Under this condition, our representative matrix will be much more original. Additionally, we obtain the submatrices of our representative matrix. Finally, we derive the most general form of our matrix.