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A {\sf $\mu$-way Latin trade} of volume $s$ is a collection of $\mu$ partialLatin squares $T_1,T_2,...,T_{\mu}$, containing exactly the same $s$ filledcells, such that if cell $(i, j)$ is filled, it contains a different entry ineach 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. %If $\mu=2$, $(T_1,T_2)$ is called a {\sf Latin bitrade}. It iscalled {\sf $\mu$-way $k$-homogeneous Latin trade}, if in each row and eachcolumn $T_r$, for $1\le r\le \mu,$ contains exactly $k$ elements, and eachelement appears in $T_r$ exactly $k$ times. It is also denoted by $(\mu,k,m)$Latin trade,where $m$ is the size of partial Latin squares.  We introduce some general constructions for $\mu$-way $k$-homogeneous Latintrades 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 areno (3,4,6) Latin trade and (3,4,7) Latin trade. Finally we present generalresults on the existence of 3-way $k$-homogeneous Latin trades for some moduloclasses of $m$.

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