Code families in McEliece cryptosytem
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.


https://binary.cr.yp.to/mcbits-20130616.pdf

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.1992.2.4.439 (paywalled, sorry!)