Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares of order $n$ can be embedded into a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. Furthermore, the constructions that we provide show that MOLS($n^2$)$\geq$MOLS($n$)+2, consequently we give a set of $9$ MOLS($576$). The maximum known size of a set of MOLS($576$) was previously given as $8$ in the literature.