Finite dynamical systems, hat games, and coding theory

TL;DR

This paper explores the properties of finite dynamical systems and their applications in coding theory and hat games, introducing new concepts like guessing and coset dimensions, and analyzing stability and instability in digraphs.

Contribution

It introduces the guessing and coset dimensions of FDSs, strengthens the link between network coding and index coding, and studies stability and instability in digraphs with new bounds.

Findings

Instability equals the size of a minimum feedback vertex set.

Certain sparse graphs with large girth have high stability and instability.

Affine instability asymptotically exceeds the linear guessing number.

Abstract

The dynamical properties of finite dynamical systems (FDSs) have been investigated in the context of coding theoretic problems, such as network coding and index coding, and in the context of hat games, such as the guessing game and Winkler's hat game. In this paper, we relate the problems mentioned above to properties of FDSs, including the number of fixed points, their stability, and their instability. We first introduce the guessing dimension and the coset dimension of an FDS and their counterparts for digraphs. Based on the coset dimension, we then strengthen the existing equivalences between network coding and index coding. We also introduce the instability of FDSs and we study the stability and the instability of digraphs. We prove that the instability always reaches the size of a minimum feedback vertex set. We also obtain some non-stable bounds independent of the number of vertices of the graph. We then relate the stability and the instability to the guessing number. We also exhibit a class of sparse graphs with large girth that have high stability and high instability; our approach is code-theoretic and uses the guessing dimension. Finally, we prove that the affine instability is always asymptotically greater than or equal to the linear guessing number.