Dutch Mathematical Olympiad

We consider security codes consisting of four digits. We say that one code dominates another code if each digit of the first code is at least as large as the corresponding digit in the second code. For example, $4961$ dominates $0761$, because $4 \ge 0$, $9 \ge 7$, $6 \ge 6$, and $1 \ge 1$. We would like to assign a colour to each security code from $0000$ to $9999$, but if one code dominates another code then the codes cannot have the same colour.
What is the minimum number of colours that we need in order to do this?