Moore determinant over a finite field

Imagine you have a big bowl of marbles. Each marble can be either red or blue. Now, let's say you want to find out if there is a way to arrange the marbles so that every row and every column of the bowl has the same number of red marbles.

To figure this out, you can use something called the Moore determinant. It's like a special tool that helps you solve puzzles like this.

First, you make a big grid that's the same size as the bowl. You put a 1 in each square of the grid if the corresponding marble in the bowl is red, and you put a 0 if it's blue.

Then, you use some fancy math to calculate a number using these 1s and 0s. This number is the Moore determinant. If the Moore determinant is not equal to 0, then you know that there is a way to arrange the marbles so that every row and every column has the same number of red marbles.

But here's the tricky part - we're not using normal numbers like 1, 2, 3, and 4 to do the math. Instead, we're using numbers from a special collection called a finite field. These numbers only go up to a certain value (called the modulus), and then they start over again at 0. It's kind of like counting on your fingers, but instead of counting up to 10 and then starting over, you only count up to 5 and then start over.

So, in summary, the Moore determinant is a tool that helps you solve puzzles by looking for patterns in a grid of numbers. And when you use it over a finite field, you're using a special set of numbers that wrap around when you count too high!