Existence of three HMOLS of type 2nu1

A Latin squares of order υ with n _i missing sub-Latin squares (holes) of order h _i (1 ≤ i ≤ k), which are disjoint and spanning (i.e. Σ _i=1 ^k n _i h _i = υ), is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by . 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(2 ⁿ 3¹) exist for n ≥ 6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS(2 ⁿ u ¹) with u ≥ 4, and prove that 3-HMOLS(2 ⁿ u ¹) exist if n ≥ 54 and n ≥ 7/4u + 7. Research supported by National Natural Science Foundation of China under Grant No. 60873267, by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y607026, and sponsored by K. C. Wong Magna Fund at Ningbo University