Given a $q$-ary code $C$ of length $n$ with minimum distance $d$, define its *rate* to be $(\log_q |C|)/n$, and its *relative distance* to be $d/n$. The Gilbert-Varshamov lower bound states that for any $q \ge 2$ and any $\delta \in (0,1)$ there is a $q$-ary code $C$ of relative distance $\ge \delta$ whose rate $r$ satisfies

$$ r \ge 1 - H_q(\delta) $$

where $H_q(\delta) = \delta \log_q(q-1) - \delta \log_q(\delta) - (1-\delta)\log_q(1-\delta)$ is the $q$-ary entropy function.

The rate of an algebraic geometry Goppa code using a curve over $\mathbb{F}_q$ of genus $g$ satisfies

$$ r \ge 1 - \delta - \frac{g-1}{n}. $$

This suggests that such codes could beat the Gilbert-Varshamov bound, and this was shown in 1982 by Tsfasman, Vladut and Zink. However I believe the best known improvement on the lower bound is very small, and so Goppa codes do not come close to meeting the Hamming bound $r \le 1-H_q(\delta/2)$.

In any case there are stronger generic bounds than the Hamming bound, for example, the Elias-Bassalygo bound, that show it is impossible to attain the channel capacity of a $q$-ary symmetric channel by hard nearest neighbour decoding in the Hamming setup.

I don't know much about decoding algebraic geometry Goppa codes. A quick web search found this paper from 1992. Roughly stated, the results in its introduction say that a Goppa code of length $n$ and minimum distance $d$ can be decoded in $O(n^3)$ time provided at most $d/2$ errors occur. There has been some more recent work on soft-decoding for Reed-Solomon codes (which are a special case of Goppa codes): see here, for example.