Let $T_i^n$ denote a particular tuple of $n$ natural numbers (here $i < n!$ and we assume that the tuple contains all elements of the set $\{0, 1, \ldots, n-2, n-1\}$, i.e. there are no duplicates). For example, $$\begin{array}{l} T_0^4 = (0,1,2,3),\\ T_1^4 = (0,1,3,2),\\ \ldots \\ T_{23}^4 = (3,2,1,0). \end{array}$$
Assuming that $(x, y)$ are two different elements of a given tuple $T$, let $d(T, x, y)$ denote the number of elements between $x$ and $y$ in $T$. For example, if $T = (3, 0, 1, 4, 2)$, we have $$d(T, 3, 0) = d(T, 1, 0) = d(T, 2, 4) = d(T, 4, 2) = 0, d(T, 3, 2) = 3.$$
Question: given an arbitrary natural number $n > 2$ and a natural number $k$ such that $0 \leq k \le n - 3$, is there an efficient algorithm to generate a set $$S = \{T_{j_0}^n, T_{j_1}^n, \ldots, T_{j_m}^n\}$$ of tuples such that for any pair $(x, y)$ of elements of the set $\{0, 1, \ldots, n-2, n-1\}$ there exists an element $T_{j_z}^n$ of $S$ such that $d(T_{j_z}^n, x, y) \leq k$, yet the cardinality of $S$ is as small as possible?
The algorithm is expected to operate as follows: start with $T_0^n$; permute $T_0^n$ to obtain $T_{j_0}^n$; permute $T_{j_0}^n$ to obtain $T_{j_1}^n$, etc. (to avoid storing a large number of tuples in memory).