The Rényi-Ulam pathological liar game with a fixed number of lies

The q-round Rényi-Ulam pathological liar game with k lies on the set [n] := {1, . . . , n} is a 2-player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [n] and Carole either assigns 1 lie to each element of A or to each element of [n]\A. Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Rényi-Ulam liar game for which the winning condition is that at most one element has k or fewer lies. We prove the existence of a winning strategy for Paul to the existence of a covering of the discrete hypercube with certain relaxed Hamming balls. Defining F ∗ k (q) to be the minimum n such that Paul can win the q-round pathological liar game with k lies and initial set [n], we find F ∗ 1 (q) and F ∗ 2 (q) exactly. For fixed k we prove that F ∗ k (q) is within an absolute constant (depending only on k) of the sphere bound, 2q/ ( q ≤k ) ; this is already known to hold for the original Rényi-Ulam liar game due to a result of J. Spencer.