Remoteness of permutation codes

Let X be a finite set of ‘points’ and d be a metric on X which takes integral values. For any v ∈ X and t ≥ 0, we refer to the set Bt(v) = {u ∈ X : d(u, v) ≤ t} as the ball of radius t centered at v, and we denote the minimum (maximum, respectively) volume of a ball with radius t in X as V min t (V max t ). Let C ⊆ X, C 6= ∅ be a code, i.e. a set of points, which we will refer to as codewords. The maximum distance between any two codewords in C is the diameter of C: δ(C) = max c,c′∈C d(c, c′),