I want to make a sequence of numbers, where I pick the numbers $a_{0}, a_{1},..,a_{n}$. The length of the sequence is $n+1$.
Now I want the product of any pair of two numbers in the sequence modulo $k$ to be guaranteed to be unique and $k$ has to be as small as it can be in such a way that the resulting set has to be $\{r_{0},r_{1},\ldots,r_{k-1\}}$. So
$$a_{0}a_{1} \bmod k \rightarrow r_{0} \\ a_{0}a_{2} \bmod k \rightarrow r_{1} \\ a_{1}a_{2} \bmod k \rightarrow r_{2} \\ \vdots\\ a_{n-1}a_{n} \bmod k \rightarrow r_{k-1}$$
First I was thinking about using prime numbers, so that the products could be unique, but I have to find a property so that every product is not congruent to every other product.
To be more general: I am looking for a sequence of length $n$ and minimal $k$ such that
$\qquad \displaystyle |\{a_ia_j \mod k \mid 0 \leq i < j \leq n\}| = n(n+1)$.
So a mapping from every product mod k to "a value" filling up every number from $0$ to $k-1$.