Non-Existence Of A Ternary Constant Weight (16,5,15;2048) Diameter Perfect Code

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 >= 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.