Families of LDPC codes with constant error fraction corrected
Here are a few examples:

There are the expander codes of Spielman (not to be confused with Sipser-Spielman):
http://www.cs.yale.edu/homes/spielman/Research/ITsuperc.pdf

There are the linear-time codes of Guruswami-Indyk:
https://www.cs.cmu.edu/~venkatg/pubs/papers/lin-zyablov.pdf

There are the expander codes of Zemor:
https://courses.engr.illinois.edu/cs598sgt/zemor.pdf