Linear Algebra Over a Finite Field

Joseph Mellor
17 min readJul 2, 2024

Let’s actually answer some questions about the cellular automata that I introduced to you a few years ago.

This article is part 9 in the series How to Discover Finite Fields While Bored in Class.

It’s been a while since I’ve published anything in this series, and I have to say that it shouldn’t have been. I’ve had this article almost ready to go for around a year. Furthermore, this is the article where I finally answer most of the questions about the cellular automata I posted about several years ago.

We’ve set up a lot of machinery throughout this series and it’s now finally time to put it to good use. By the end of this article, we’ll get a reasonably fast algorithm for determining the both the cycle length and the presence of a delay that works in 80% of all possible cases. We still have a long ways to go, but we’ll get there.

Prerequisites

While this is part of a series, you don’t need to know anything from the previous articles except:

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Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.