Unsolved Perfect Coverings

Question

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?

Answer 1

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.