generalized MRRW bound on the asymptotic rate of q-ary codes Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ corresponding to a relative distance of $\delta$ is such that:
\begin{equation*}R(\delta) \leq H_2(\frac{1}{2}-\sqrt{\delta(1-\delta)}) \end{equation*}
where H is the (binary) entropy function.
(A slight improvement of the above exists in the binary case, but within the same framework)
In the case of q-ary codes, i.e. codes over $\mathbb F _q ^n$, the above bound is generalized to:
\begin{equation*}R(\delta) \leq H_q(\frac{1}{q}(q-1-(q-2)\delta-2\sqrt{(q-1)\delta(1-\delta)})) \end{equation*}
My question is as follows:
For larger alphabet size q, the above bound seems to weaken significantly. In fact, observing the growth of the above bound as $q \rightarrow \infty$ (using simple approximations of entropy), we see that:
\begin{equation*} R(\delta) \leq 1-\delta+\mathcal{O}(\frac{1}{\log{q}})  \end{equation*}
Thus, it seems to get worse than even the Singleton bound $R(\delta) \leq 1-\delta$.
So which is the best bound for large alphabet size $q$? Or am I wrong in the above conclusion (most sources claim the MRRW bound stated above is the best known bound, but its not clear if that holds for larger q as well).
Also, could someone direct me to references for comparisons of different bounds for larger $q$? I am able to find reliable comparisons only for $q=2$.

The source is formula (3) on page 86 in the artice
Aaltonen, Matti J.: Linear programming bounds for tree codes. IEEE Transactions on Information Theory 25.1 (1979), 85–90,
doi: 10.1109/tit.1979.1056004.
According to the article, there is the additional requirement $0 < \delta < 1 - \frac{1}{q}$.
Instead of comparing to the asymptotic Singleton bound, one can (more ambitiously) compare to an improvement, the asymptotic Plotkin bound
$$R(\delta) \leq 1 - \frac{q}{q-1}\,\delta$$
for $0 \leq \delta \leq q - \frac{1}{q}$.
In Aaltonen's paper, the formula is followed by some discussion into this direction. There I find "However, for large $q$ the Plotkin bound remains the best upper bound."
So apparently this "generalized MRRW bound" is not strong for large $q$, as you say.
As a side note, for $q\to \infty$, the function $R(\delta)$ converges to the asymptotic Singleton bound $1 - \delta$.
