What are the families of codes frequently used in McEliece cryptosystem or its variants? I know that binary Goppa codes were used in the original system but many codes with efficient decoding algorithm can be used. Are there any other requirements on the code or generator matrix for that matter? I don't care about parameters used, just the names of families like Goppa, Hamming , Reed Solomon etc. is enough.
The modern approach is still to use binary Goppa codes. See, e.g., McBits from 2013:
Daniel J. Bernstein, Tung Chou, Peter Schwabe. "McBits: fast constant-time code-based cryptography." Pages 250–272 in Cryptographic hardware and embedded systems—CHES 2013—15th international workshop, Santa Barbara, CA, USA, August 20–23, 2013, proceedings, edited by Guido Bertoni, Jean-Sébastien Coron. Lecture Notes in Computer Science 8086, Springer, 2013. ISBN 978-3-642-40348-4.
Other proposals, such as Niederreiter's using generalized Reed–Solomon codes, have been broken:
V.M. Sidelnikov, S.O. Shestakov. "On insecurity of cryptosystems based on generalized Reed–Solomon codes." Discrete Mathematics and Applications, 2(4), pp. 439–444, 1992.
https://doi.org/10.1515/dma.1922.214.171.1249 (paywalled, sorry!)