Perfect, minimally adaptive, error-correcting searching strategies

Let q_e(m) be the smallest integer q satisfying Berlekamp´s bound Σ_i=0^e(_i^q )⩽2^q-m. We prove that for any fixed e⩾1 and all sufficiently large m there is a binary searching strategy to guess a number x∈{0,...,2^m-1} in spite of up to e lies in the answers, which uses exactly q_e(m) questions and adaptiveness only once. The strategy goes through a first batch of m non-adaptive questions asking for the bits of the binary expansion of x and then, only depending on the answers to these questions, a second batch of q _e(m)-m non-adaptive questions