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

A New Bound for the Orthogonality Defect of HKZ Reduced Lattices