2

I'm trying to learn to use MAGMA(Computational Algebra System) for research in coding theory over non-commutative rings, but it's been slow going. I feel like it's hard to find anything in the handbook.

For example, I need to define a non-commutative ring $R:=\mathbb{F}_{4}+e\mathbb{F}_{4}$, where $\mathbb{F}_{4}$ is the finite field of $4$ elements and $e^2=1$, $ae=ea^2$. But I didn't find the information to help me.

I would like to take your precious time, please help me to write the code of this non-commutative ring in MAGMA. I know the best way to learn programming about MAGMA is to read the code, but I don't know where to get them. Could you give me some suggestions?

Thanks a lot for your answer and best wishes!

Xuesong Si
  • 85

1 Answers1

3

By the way, there is this question on the site which might interest you in general: Learning to use MAGMA

I think what you are looking for is a quotient of a twisted polynomial ring

$$F_4[x;\sigma]/(x^2-1)$$

where $\sigma: a\mapsto a^2$ is the Frobenius isomorphism on $F_4$.

Googling 'twisted polynomial magma cas' got me to this documentation.

Now I do not have MAGMA installed and I can't test to make sure thse programmers have implemented "twisted polynomials" as ring theorists call them, but they do reference the Frobenius endomorphism on that page, so I think it's highly likely to be what ring theorists call "twisted polynomials."

If all goes well it will tell you how to create $F_4[x;\sigma]$ and then hopefully it supports creation of quotient by the ideal $(x^2-1)$ as well.

azimut
  • 24,316
rschwieb
  • 160,592