Asymptotically good codes
Usually the question is interesting for constant alphabet sizes, since otherwise Reed-Solomon codes obviously achieve the Singleton bound.

For constant (but still large) alphabet sizes, there are explicit codes that "approximately" achieve the Singleton bound, and are thus also asymptotically good. See

Venkatesan Guruswami and Piotr Indyk. Linear time encodable and list decodable codes Proceedings of STOC 2003.

Also there are several flavors of Expander and Concatenated Codes (rather than the famous constructions) that are asymptotically good. For example see

"Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs" by Alon et al.

Also there are more sophisticated "concatenated-like" constructions that achieve much more than asymptotic-goodness, for example:

V. Guruswami and A. Smith. Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate, FOCS 2010.

Finally, any construction of codes for stochastic errors (that is, the channel BSC(p)) that achieves constant rate and exponentially small error must be asymptotically good. However, I'm not aware of any such explicit constructions other than concatenated codes (Forney/Justesen and similar variations).