A Kraft-Type Inequality for d-Delay Binary Search Codes

Among the models of delayed search discussed in [1], [2], the simplest one can be formulated as the following two–player game. One player, say A, holds a secret number m ∈ M {1, 2, . . . ,M} and another player, say Q, tries to learn the secret number by asking A at time i questions, like “Is m ≥ xi?”, where xi is a number chosen by Q. The rule is that at time i+ d A must answer Q’s question at time i correctly and at time j Q can choose xj according to all answers he has received. How many questions has Q at least to ask to get the secret number. Let