  In this paper, we show that for each lattice basis, there exists an
equivalent basis which we describe as ``strongly reduced''. We show that bases
reduced in this manner exhibit rather ``short'' basis vectors, that is, the
length of the $i$th basis vector of a strongly reduced basis is upper bounded
by a polynomial factor in $i$ multiplied by the $i$th successive minima of the
lattice. The polynomial factor seems to be smaller than other known factors in
literature, such as HKZ and Minkowski reduced bases. Finally, we show that such
bases also exhibit relatively small orthogonality defects.
