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latin square; latin trade; abelian $2$-group
Let $T$ be a partial latin square and $L$ be a latin square with $T\subseteq L$. We say that $T$ is a latin trade if there exists a partial latin square $T'$ with $T'\cap T=\emptyset $ such that $(L\setminus T)\cup T'$ is a latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry either $0$ or $k$ times. In this paper, we show the existence of $3$-homogeneous latin trades in abelian $2$-groups.
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