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

Abstract: A {\sf $\mu$-way Latin trade} of volume $s$ is a collection of $\mu$ partial Latin squares $T_1,T_2,...,T_{\mu}$, 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. %If $\mu=2$, $(T_1,T_2)$ is called a {\sf Latin bitrade}. It is called {\sf $\mu$-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 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 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 are no (3,4,6) Latin trade and (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 $m$.