How flexibible can one choose the properties of algebraic-geometric codes?

Question

Theorem 7.3 on page 67 in these lecture notes states the following.

For every even power of a prime $q$, and every parameter $\delta < 1 - 1/(\sqrt{q} - 1)$, there exists an infinite family of $q$-ary linear codes of relative distane $\delta$ and rate $R \geq 1- \delta - 1/(\sqrt{q} - 1)$. Further a generator matrix for such a code can be construted in $\tilde{\mathcal{O}}(n^3)$ time, where $n$ is the length of the code.

While this is really nice in asymptotic contexts, I wonder how flexible one can choose the concrete parameters of these algebraic geometric codes. For example: Could one choose $q = 64, \delta = 0.19$ and $k = 2$ to have an $[3, 2, 1]_64$ code? Or phrased more generally: Which constraints are there on the choice of these parameters?

I tried reading the paper the theorem above is based on, but I am to unacquainted to algebraic geometry to answer the question myself.

Answer 1

There is no single theorem that says what parameters are possible.

If you Ctrl-F (or whatever equivalent) your document for “bound”, you will find several famous constrains on the relationship between code length, alphabet size and distance. That is what you are looking for, essentially.

In coding theory, you will encounter many families of codes, all of which have different properties and performance. All you can do is check their parameters. I don’t think it’s feasible to exhaustively search for combinations of parameters.

Answer 2

This question was answered to some satisfaction in the comment by Jyrki Lahtonen, as written by the poster in another comment.

From the comments:

"Those results (at least most of them) are asymptotic in nature. To realize a fixed pair of parameters $(R,\delta)$ may require quite long codes. I'm afraid I'm not up to speed with what exactly is possible. I would look up Garcia-Stichtenoth curves (or a tower of curves/function field extensions) to get a more precise idea about explicit parameter sets. I'm not sure whether the original Tsfasman-Vladuts-Zink results were based on explicit curves or simply on some existence results. I would need to check a few sources to say anything precise."