Existence of Frame SOLS of Type $3^n{u}^1$

Abstract

An SOLS (self-orthogonal Latin square) of order $n$ with $n_i$ missing sub-SOLS (holes) of order $h_i$ ($1\le i\le k$), which are disjoint and spanning (i.e., $\sum _{i=1}^k{n}_i{h}_i=n$), is called a frame SOLS and denoted by $\text{FSOLS}(h_1^{n_1},h_2^{n_2},\dots ,h_k^{n_k})$.In this article, it is shown that an $\text{FSOLS}(3^{n-u}{3}^1)$ exists if and only if $n\ge 4$ and $n\ge 1+\frac{2u}{3}$, with seventeen possible exceptions $(n,u)=\{(5,1)\}$ and $\{(n,u)=(n,\lfloor \frac{3(n-1)}{2}\rfloor )$ for $(n\in \{6,10,14,18,22,30,34,38,42,46,54,58,62,66,70,94\}$.