Let $k$ and $\ell$ be positive integers and define $[k] = \{ 1,\dots,k\}$.
Suppose we are given a set $S$ of $\ell$-length sequences, each one of the form $(s_1,\dots,s_\ell)$, where each $s_i \in [k]$. There are no other restrictions on the elements in $S$.
What size does $S$ need to have such that, for at least one index $j \le \ell$, every $s \in [k]$ occurs in some sequence in $S$ at index $j$?
The obvious upper bound is of course $k^\ell$. Is it possible to show a tighter bound or is this as good as it gets?
Consider the lists in $S$ arranged as rows of a table, with column $i$ containing all the $i$th terms of the sequences. We want to see how big we can make $S$ such that no column has every number.
The largest size $S$ will satisfy that every column will have $k-1$ different numbers because if not, then some column $j$ has less than $k-1$ different numbers we could add another sequence to $S$ that is identical to an existing one but with the $j$th term changed to one of the numbers that were not present in column $j$. Then we still have that no column contains all $k$ numbers so we are fine.
It should be noted that in the table, numbers may appear more than once in a column, but we are only interested in how many different numbers appear in the column.
It doesn't matter what the $k-1$ different numbers are, since the largest $S$ will contain all the combinations of each selection of $k-1$ numbers from each column.
Hence the we only need $S$ to be at least size $(k-1)^l+1$