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.

翻译：本文证明，对于每个格基，存在一个我们称之为“强约化”的等价基。我们表明，以这种方式约化的基具有较“短”的基向量，即强约化基的第i个基向量的长度被格的第i个逐次最小值的多项式倍数所上界。该多项式因子似乎小于文献中已知的其他因子，例如HKZ和Minkowski约化基的因子。最后，我们证明此类基还表现出相对较小的正交性缺陷。