Existence of three HMOLS of type 2n u 1

A Latin squares of order . with n i missing sub-Latin squares (holes) of order h i (1 . i . k), which are disjoint and spanning (i.e. . k i=1 n i h i = .), is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by (hn11hn22.hnkk). If any two PILS in a set of t PILS of type T are orthogonal, then we denote the set by t-HMOLS(T). It has been proved that 3-HMOLS(2n31) exist for n . 6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS(2n u 1) with u . 4, and prove that 3-HMOLS(2n u 1) exist if n . 54 and n . 7/4u + 7.