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On the Dynamics of Linear Finite Dynamical Systems Over Galois Rings

Jonas Kantic, Claudio Qureshi, Daniel Panario, Fabian Legl

Abstract

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.

On the Dynamics of Linear Finite Dynamical Systems Over Galois Rings

Abstract

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.

Paper Structure

This paper contains 17 sections, 32 theorems, 82 equations, 1 figure, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Background in Abstract Algebra
  4. Canonical Homomorphism
  5. Factorization of Polynomials
  6. Finite Dynamical Systems
  7. Linear Finite Dynamical Systems
  8. LFDSs over Galois fields
  9. LFDSs over Galois rings
  10. LFDSs on Cyclic Modules Over Finite Fields
  11. Related Work
  12. LFDSs on Cyclic Modules Over Galois Rings
  13. Bijective LFDSs on Cyclic Modules Over Galois Rings
  14. Algorithm to Compute the Set of Cycle Lengths
  15. Reducing the Number of Factors
  16. ...and 2 more sections

Key Result

Lemma 1

Let $S$ be a ring and let $g \in S\mleft[ x \mright]$ be a monic polynomial. If $S$ is torsion-free, then $S[x]/\langle g \rangle$ is also torsion-free. In particular, $R = \mathbb{Z}[t]/\langle h \rangle$ is torsion-free. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Number of irreducible divisors $G_i$ of $M_1(x)$ from \ref{['exp:many-factors']}, aggregated per order.

Theorems & Definitions (67)

  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 1
  • Lemma 4
  • Definition 2
  • ...and 57 more