As an example I describe what is known about the BCH-codes for the purposes of this question. Just to give an idea of the type of results that are known. I suspect that they are not very useful for the full range of parameters here.
Let us specify a natural number $m$, and let $n=2^m-1$. Let us further specify the parameter $t$ (=the number of bit errors the code can correct). The BCH-code (more accurately, a narrow sense, primitive BCH-code) with designed minimum distance $d_{des}=2t+1$ is a vector space $V(m,t)\subset \Bbb{F}_2^n$ has minimum Hamming distance $d_{min}\ge d_{des}$ and dimension
$$k(m,t)=\dim V(m,t)\ge n-mt.$$
The other parameter of interest here is the covering radius $\rho(m,t)$ of $V(m,t)$. That is, the smallest integer $\rho$ such that every point of $\Bbb{F}_2^n$ is within Hamming distance $\rho$ of a vector from the subspace $V(m,t)$.
Quite a bit is known about $\rho(m,t)$. This revolves around the solvability of systems of equations over $\Bbb{F}_{2^m}$.
- In 1987 Tietäväinen proved that $\rho(m,t)\le 2t$ whenever $(2t-1)^{4t+2}\le 2^m$. This places a rather high demand on $m$, and that may prove to be the downfall of the use of BCH-codes here.
- In most cases (and also asymptotically) the true covering radius is $\rho=2t-1$, a result of Vladuts & Skorobogatov. The length from which on this holds has been brought down by a number of authors since.
- Usually $\rho\ge2t-1$ follows from the so called supercode lemma. That is easy to describe. We always have $V(m,t)\subseteq V(m,t-1)$. Whenever the inclusion here is strict, and the code $V(m,t-1)$ contains a vector $x$ of Hamming weight $2t-1$ (both of these are true often enough), then $x$ obviously cannot be at a distance $<2t-1$ from a vector of $V(m,t)$, proving the claim.
- A bit of searching pointed me to a paper by F. Levy-dit-Vehel and S. Litsyn that appeared in IEEE Transactions on Information Theory, vol. 42(3) in May '96. They described the known results at that time.
- So for BCH-codes the covering radius is often relatively close to the minimum distance (if it were higher, we could make the code bigger). Meaning that your parameter $\delta$ can be made close to $1$.
- The catch is that the BCH-codes are known to be "asymptotically bad". From the inequality of the dimension we see that $t$ is usually bounded to be at most something like $n/\log_2n$. Also the bounds on $\rho$ don't apply to very small BCH-codes.
Another family of codes whose covering radii have been studied extensively consists of the Reed-Muller codes. The first order Reed-Muller codes in particular could be useful for your purpose.
The main reason you may not have found a lot of information is that for coding theoretical purposes, whenever $\rho>d_{min}$ the code becomes immediately non-interesting. I guess that's the content of Ilya Bogdanov's answer in different language.
