The Existence of Four HMOLS with Equal Sized Holes

In this paper, we present several new constructions for k holey mutually orthogonal Latin squares (HMOLS) of type gn. We concentrate mainly on k=4; here, for all but two values of n, namely 6 and 15, only a finite number of unsolved cases remain. Some new sets of 5 and 6 HMOLS are also given, in particular 5 HMOLS(2q) for q≥63 or q an odd prime power between 6 and 62, plus 6 HMOLS(4q)for q an odd prime power between 8 and 60.

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