On the Dynamics of Linear Finite Dynamical Systems Over Galois Rings
This work addresses a problem in coding theory and simulations by providing practical tools for analyzing system dynamics, though it appears incremental as it extends existing analysis to a broader algebraic structure.
The paper tackles the analysis of linear finite dynamical systems over Galois rings, extending methods from cyclic modules to compute cycle lengths and tree heights in functional graphs, with algorithms proposed for these computations.
Linear finite dynamical systems play an important role, for example, in coding theory and simulations. Methods for analyzing such systems are often restricted to cases in which the system is defined over a field %and usually strive to achieve a complete description of the system and its dynamics. or lack practicability to effectively analyze the system's dynamical behavior. However, when analyzing and prototyping finite dynamical systems, it is often desirable to quickly obtain basic information such as the length of cycles and transients that appear in its dynamics, which is reflected in the structure of the connected components of the corresponding functional graphs. In this paper, we extend the analysis of the dynamics of linear finite dynamical systems that act over cyclic modules to Galois rings. Furthermore, we propose algorithms for computing the length of the cycles and the height of the trees that make up their functional graphs.