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

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.

• 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. Dec 30 '18 at 20:03
• @JyrkiLahtonen Thank you for your comment. Something like this was what I was looking for. Concerning the explicitness, I think it was the way that Garcia and Strichtenloh were the first to provide an polynomial time construction. However, the polynomial was something like $n^30$. Then the paper I linked above was the first to come up with a construction in time $\tilde{\mathcal{O}}(n^3)$.
– Dave
Dec 30 '18 at 21:42

• Thank you for your quick response. I already know about the several bounds. What I wonder about much more is the granularity in the concrete case. Meaning if it is possible to construct codes for all $n$ in the bounds or if there for example has to exist a set with certain properties which does not always exist.