  In this work an efficient algorithm to perform a block decomposition (and so
to compute the rank) of large dense rectangular matrices with entries in
$\mathbb{F}_2$ is presented. Depending on the way the matrix is stored, the
operations acting on rows or block of consecutive columns (stored as one
integer) should be preferred. In this paper, an algorithm that completely
avoids the column permutations is given. In particular, a block decomposition
is presented and its running times are compared with the ones adopted into
SAGE.
