Existence of frame SOLS of type ab

An SOLS (self-orthogonal latin square) of order v with n_i missing sub-SOLS (holes) of order h_i (1ik), which are disjoint and spanning (i.e. ∑_i=1^k n_ih_i=v), is called a frame SOLS and denoted by FSOLS(h₁^n₁h₂^n₂ h_k^n_k). It has been proved that for b2 and n odd, an FSOLS(aⁿb¹) exists if and only if n4 and n1+2b/a. In this paper, we show the existence of FSOLS(aⁿb¹) for n even and FSOLS(aⁿ1¹) for n odd.