Existence of $(3,1,2)$-HCOLS and $(3,1,2)$-ICOILS

Abstract

A Latin square $(S,\cdot )$ is said to be $(3,1,2)$-$conjugate-orthogonal$ if $x\cdot y=z\cdot w$, $x{\cdot }_{312}y=z{\cdot }_{312}w$ imply $x=z$ and $y=w$, for all $x,y,z,w\in S$, where $x_3{\cdot }_{312}{x}_1=x_2$ if and only if $x_1\cdot x_2=x_3$.Such a Latin square is said to be $holy$ $((3,1,2)$-HCOLS for short) if it has disjoint and spanning holes corresponding to missing sub-Latin squares.Let $(3,1,2)$-HCOLS$(h^n)$ denote a $(3,1,2)$-HCOLS of order $hn$ with $n$ holes of equal size $h$.We show that, for any $h\ge 1$, a $(3,1,2)$-HCOLS$(h^n)$ exists if and only if $n\ge 4$, except $(n,h)=(6,1)$, and except possibly $(n,h)=(10,1)$ and $(4,2t+1)$ for $t\ge 1$.Let $(3,1,2)$-ICOILS$(v,k)$ denote an idempotent $(3,1,2)$-COLS of order $v$ with a hole of size $k$.We prove that a $(3,1,2)$-ICOILS$(v,k)$ exists for all $v\ge 3k+1$ and $1\le k\le 5$, except possibly $k=4$ and $v\in \{35,38\}$.