Colin Defant - Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

dmtcs:2654 - Discrete Mathematics & Theoretical Computer Science, October 3, 2017, Vol. 19 no. 3 - https://doi.org/10.23638/DMTCS-19-3-10
Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

Authors: Colin Defant

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.

Volume: Vol. 19 no. 3
Section: Combinatorics
Published on: October 3, 2017
Submitted on: May 30, 2017
Keywords: Mathematics - Combinatorics,Computer Science - Information Theory,37E15, 05C69