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Let $T$ be a partial latin square and $L$ a latin square such that
$T\subset L$. Then $T$ is called a {\it latin trade}, if there
exists a partial latin square $T^*$ such that $T^* \cap T = \phi$
and $(L \setminus T)\cup T^*$ is a latin square. We call $T^*$ a
{\it disjoint mate} of $T$. A latin trade is called {\it
k-homogeneous} if each row and each column contains exactly $k$
elements, and each element appears exactly $k$ times. The number
of elements in a latin trade is referred to as its {\it volume}.
It is shown by Cavenagh, Donovan, and Drapal (2003 and 2004) that
3-homogeneus and 4-homogeneous latin trades of volume $3m$ and
$4m$, respectively, exist for all $m\geq 3$ and $m\geq 4$,
respectively. We show that k-homogeneous latin trades of volume
$km$ exist for all $3\leq k \leq 8$ and $m \geq k$. Also we show
that for each given $k\geq 3$ and $m\geq k$, all k-homogeneous
latin trades of volume $km$ exist except possibly for finitely
many $m$, i.e. $k