In this question about perfect 2 error correcting codes on the Open Problem Garden, it is stated that:

Recent research activity has discovered a large number of previously unknown perfect 1-error correcting codes which are not isomorphic to the Hamming codes.

In terms of context, the following is well known for $t-$error correcting codes over finite alphabets:

No perfect $t-$error correcting codes other than the repetition codes (any $t\geq 1,$ any alphabet, odd length), the Hamming codes (binary alphabet, $t=1$) and the two Golay codes (binary with length $n=23$ and $t=3$ and ternary with length $n=11,$ and $t=2$) exist in the Hamming metric when the alphabet size $q$ is a prime power.

**Question:** *Does anyone here know what these (recently found circa 2010) codes are?*

The below is the stated open question for $t=2,$ but to be a conjecture it must actually say there are or there aren't such codes:

Conjecture: Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?