Strongly Reduced Lattice Bases

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 rathershort'' 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.