  Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality
$2^n$ and distance $5$ are known to exist for every $m$ for which there exists
an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and
for $m=6$. We show the non-existence of such codes for $m=4$ and prove that any
codes with the parameters above are diameter perfect.
