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We provide the first hardness result for the Covering Radius Problem on lattices (CRP). Namely, we show that for any large enough p ≤ ∞ there exists a constant c p > 1 such that CRP in the ℓ p norm is Π 2 -hard to approximate to within any constant factor less than c p . In particular, for the case p = ∞, we obtain the constant c ∞ = 3/2. This gets close to the factor 2 beyond which the problem is not believed to be Π 2 -hard (Guruswami et al., Computational Complexity, 2005).doi:10.1109/ccc.2006.23 dblp:conf/coco/HavivR06 fatcat:yhnu35aj7vdx5o5dfsbefsnwg4