Improvements for lower bounds of mutually orthogonal Latin squares of sizes 54, 96 and 108

Abstract In this paper, respectively 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of sizes $$n=54$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>54</mml:mn> </mml:mrow> </mml:math> , 96 and 108 are obtained (previously, only 5, 9 and 8 MOLS were respectively known for these values). The cases $$n=54$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>54</mml:mn> </mml:mrow> </mml:math> and 96 are obtained by constructing separable permutation codes consisting of $$8 \times 54$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> <mml:mn>54</mml:mn> </mml:mrow> </mml:math> and $$10 \times 96$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>10</mml:mn> <mml:mo>×</mml:mo> <mml:mn>96</mml:mn> </mml:mrow> </mml:math> 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 $$n=35$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>35</mml:mn> </mml:mrow> </mml:math> , 54 and 63, respectively. The case $$n=108$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>108</mml:mn> </mml:mrow> </mml:math> is obtained by constructing a (108, 10, 1) difference matrix. Also, a previous error when constructing 6 MOLS of size $$n=45$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>45</mml:mn> </mml:mrow> </mml:math> is corrected.