The Orthomorphism Graph $\mathcal{L}_3(q)$

Abstract

Orthomorphisms of groups and pairwise orthogonal orthomorphisms have been used in several constructions of combinatorial designs, in particular in the construction of mutually orthogonal sets of latin squares based on groups. In this paper we will use difference equations to construct orthomorphisms in $\mathcal{L}_3(q)$, an orthomorphism graph of $GF(q)^{+}\times GF(3)^+$, and to establish lower bounds for the number of pairwise orthogonal orthomorphisms in $\mathcal{L}_3(q)$.