Existence of frame SOLS of type anb1

AbstractAn SOLS (self-orthogonal latin square) of order v with ni missing sub-SOLS (holes) of order hi (1⩽i⩽k), which are disjoint and spanning (i.e. ∑i=1knihi=v), is called a frame SOLS and denoted by FSOLS(h1n1h2n2 ⋯hknk). It has been proved that for b⩾2 and n odd, an FSOLS(anb1) exists if and only if n⩾4 and n⩾1+2b/a. In this paper, we show the existence of FSOLS(anb1) for n even and FSOLS(an11) for n odd