Covering radius in the Hamming permutation space

Let S_n 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⊆S_n with the property that for any ρ∈S_n 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 the whole conjecture. We also show that f(n,2)>3n∕4 for all n, that s!<f(n,s)<3s!(n−s)logn for 1⩽s⩽n−2 and that f(n,s)> [Formula presented] [Formula presented] if s⩾3.