On the existence of 3-way k-homogeneous Latin trades

A $\mu$-way Latin trade of volume $s$ is a collection of $\mu$ partial Latin squares $T_1,T_2,\dots ,T_{\mu }$, containing exactly the same $s$ filled cells, such that, if cell $(i,j)$ is filled, it contains a different entry in each of the $\mu$ partial Latin squares, and such that row $i$ in each of the $\mu$ partial Latin squares contains, set-wise, the same symbols, and column $j$ likewise. It is called a $\mu$-way$k$-homogeneous Latin trade if, in each row and each column, $T_r$, for $1\le r\le \mu$, contains exactly $k$ elements, and each element appears in $T_r$ exactly $k$ times. It is also denoted as a $(\mu ,k,m)$ Latin trade, where $m$ is the size of the partial Latin squares.We introduce some general constructions for $\mu$-way $k$-homogeneous Latin trades, and specifically show that, for all $k\le m$, $6\le k\le 13$, and $k=15$, and for all $k\le m$, $k=4,5$ (except for four specific values), a $3$-way $k$-homogeneous Latin trade of volume $km$ exists. We also show that there is no $(3,4,6)$ Latin trade and there is no $(3,4,7)$ Latin trade. Finally, we present general results on the existence of $3$-way $k$-homogeneous Latin trades for some modulo classes of $m$.