Covering radius in the Hamming permutation space

Let Sn denote the set of permutations of {1,2,…,n}. The function f(n,s) is defined to be the minimum size of a subset S⊆Sn with the property that for any ρ∈Sn there exists some σ∈S such that the Hamming distance between ρ and σ is at most n−s. The value of f(n,2) is the subject of a conjecture by Kézdy and Snevily, which implies several famous conjectures about Latin squares. We prove that the odd n case of the Kézdy–Snevily Conjecture implies...