Unsolved Perfect Coverings I went through the La Jolla Covering Repository and wrote about perfect coverings in a community post. 
UNSOLVED:
C(36,6,2), C(46,6,2), C(51,6,2),
C(61,6,2), C(81,6,2), C(43,7,2),
C(85,7,2), C(37,7,3), C(42,6,4)  
The above are the cases where a perfect covering hasn't been proven impossible, but a solution has not yet been found.  I was especially surprised by C(43,7,2) not being solved.  
Is there a list somewhere of unsolved combinatorial problems of this sort?  Are any of these cases vulnerable to computer attacks or impossibility proofs?

Assuming that I'm correct to interpret your "perfect coverings" as Steiner systems, there's a list of known status for possible parameters elsewhere in the La Jolla Covering Repository: https://www.ccrwest.org/cover/steiner.html
Note in particular:

36    6   2       Does not exist (see Colbourn and Mathon)
  43    7   2       Does not exist (see Colbourn and Mathon)
  37    7   3       Does not exist (see Colbourn and Mathon)  

The reference is to Steiner Systems, Charles J. Colbourn and Rudolf Mathon, in Handbook of Combinatorial Designs, second edition, (2007) pp. 102-110.
