Latin bitrades derived from quasigroup autoparatopisms

TL;DR

This paper generalizes the construction of Latin trades from group-based methods to embed them directly into Latin squares using quasigroup autoparatopisms, enabling the identification of non-trivial entry-transitive trades.

Contribution

It introduces a new approach to embed Latin trades into Latin squares via quasigroup autoparatopisms, extending previous group-based constructions.

Findings

Identified non-trivial entry-transitive trades in group operation tables

Constructed Latin trades embedded in specific Latin squares

Applied theory to quadratic orthomorphisms

Abstract

In 2008, Cavenagh and Dr\'{a}pal, et al, described a method of constructing Latin trades using groups. The Latin trades that arise from this construction are entry-transitive (that is, there always exists an autoparatopism of the Latin trade mapping any ordered triple to any other ordered triple). Moreover, useful properties of the Latin trade can be established using properties of the group. However, the construction does not give a direct embedding of the Latin trade into any particular Latin square. In this paper, we generalize the above to construct Latin trades embedded in a Latin square $L$, via the autoparatopism group of the quasigroup with Cayley table $L$. We apply this theory to identify non-trivial entry-transitive trades in some group operation tables as well as in Latin squares that arise from quadratic orthomorphisms