The existence of four HMOLS with equal sized holes (Q1611352)

The authors present several new constructions for \(k\) holey mutually orthogonal Latin squares (HMOLS) of type \(g^n\). They concentrate mainly on the case \(k = 4\), and here they show that, for all but two values of \(n\), namely \(6\) and \(15\), only a finite number of unsolved cases remain. They also present some new sets of \(5\) and \(6\) HMOLS, in particular 5 HMOLS(\(2^q\)) for \(q \geq 63\) or \(q\) an odd prime power between \(6\) and \(62\), plus 6 HMOLS(\(4^q\)) for \(q\) an odd prime power between \(8\) and \(60\). For \(2 \leq g \leq 6\) and \(n < 60\) they provide a number of new direct constructions.