Linear Algebra Over a Finite Field

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.

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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:

• An Abelian group is a set of objects that acts like the integers under addition.
• A field is a set of objects you can add, subtract, multiply, and divide just like the real numbers.
• In modular arithmetic, you add, subtract, and multiply the integers as you…