Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

TL;DR

This paper links the maximum number of period-2 orbits in a specific class of sequential dynamical systems to the maximum size of certain binary codes, revealing a new interpretation of a known coding sequence.

Contribution

It establishes a novel connection between dynamical systems and coding theory by interpreting the maximum number of period-2 orbits as a binary code problem.

Findings

ta_n equals the maximum size of a binary code with minimum distance 3.

First interpretation of this coding sequence in a dynamical systems context.

Provides a new perspective on the relationship between SDS maps and coding theory.

Abstract

Let $[K_n,f,\pi]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_n$ using the update order $\pi\in S_n$ in which all vertex functions are equal to the same function $f\colon\mathbb F_2^n\to\mathbb F_2^n$. Let $\eta_n$ denote the maximum number of periodic orbits of period $2$ that an SDS map of the form $[K_n,f,\pi]$ can have. We show that $\eta_n$ is equal to the maximum number of codewords in a binary code of length $n-1$ with minimum distance at least $3$. This result is significant because it represents the first interpretation of this fascinating coding-theoretic sequence other than its original definition.