Non-existence of a ternary constant weight (16, 5, 15; 2048) diameter perfect code

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Ternary constant weight codes of length n=2ᵐ, weight n-1, cardinality 2ⁿ and distance 5 are known to exist for every m for which there exists an APN permutation of order 2ᵐ, 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.