The Spectrum for $3$-Way $k$-Homogeneous Latin Trades

Abstract

A $\mu$-way $k$-homogeneous Latin trade was defined by Bagheri Gh, Donovan, Mahmoodian (2012), where the existence of $3$-way $k$-homogeneous Latin trades was specifically investigated. We investigate the existence of a certain class of $\mu$-way $k$-homogeneous Latin trades with an idempotent like property. We present a number of constructions for $\mu$-way $k$-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of $3$-way $k$-homogeneous Latin trades for all but $196$ possible exceptions.