A New Bound for the Orthogonality Defect of HKZ Reduced Lattices

In this work, we determine a sharp upper bound on the orthogonality defect of HKZ reduced bases up to dimension $3$. Using this result, we determine a general upper bound for the orthogonality defect of HKZ reduced bases of arbitrary rank. This upper bound seems to be sharper than existing bounds in literature, such as the one determined by Lagarias, Lenstra and Schnorr.