Create $n$ subset of $S$ so that union of any $k$ of them is equal to $S$

Question

Given a set S, how to create n subset S₁, ... S_n of S, so that union of any k of them is equal to S but no k-1 unions equals to S.

Example:

S = {0,1,2,3,4,5,6,7,8,9}
n = 5
k = 3
S1 = {0, 1, 2, 3, 4, 5}
S2 = {0, 1, 2, 6, 7, 8}
S3 = {0, 3, 4, 6, 7, 9}
S4 = {1, 3, 5, 6, 8, 9}
S5 = {2, 4, 5, 7, 8, 9}

|S|=10
1<=n<=9
1<=k<=n

I have observed that, every element in S is repeated n-k+1 times. But stuck there.
Any useful thoughts are appreciated.