  Given two integers $m$ and $n$ with $m\leq n$, a Latin rectangle of size
$m\times n$ is a bi-dimensional array with $m$ rows and $n$ columns filled with
symbols from an alphabet with $n$ symbols, such that each row contains a
permutation of the alphabet and each column contains no repeated symbols.
  Two rows $a$ and $b$ of a Latin rectangle $R$ define a permutation $R_{a,b}$
assigning the symbol $y$ to the symbol $x$ if they are in the same column, $x$
is in row $a$ and $y$ is in row $b$. A Latin rectangle $R$ is perfect is the
permutation $R_{a,b}$ is cyclic, for each pair of rows $a$ and $b$.
  We prove that for each integer $m$ and each large enough odd integer $n$
there is a perfect Latin rectangle $R$ of size $m\times n$. It is a partial
(asymptotic) answer to a well-known conjecture which says that the same
property holds for each odd integer $m\leq n$.
