Abstract In this paper, respectively 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of sizes $$n=54$$ n = 54 , 96 and 108 are obtained (previously, only 5, 9 and 8 MOLS were respectively known for these values). The cases $$n=54$$ n = 54 and 96 are obtained by constructing separable permutation codes consisting of $$8 \times 54$$ 8 × 54 and $$10 \times 96$$ 10 × 96 codewords, respectively; in addition, these codes respectively have lengths 54, 96 and minimum distances 53, 95. This construction method was used in an earlier paper to obtain 6, 10 and 8 MOLS of sizes $$n=35$$ n = 35 , 54 and 63, respectively. The case $$n=108$$ n = 108 is obtained by constructing a (108, 10, 1) difference matrix. Also, a previous error when constructing 6 MOLS of size $$n=45$$ n = 45 is corrected.