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ADAPTIVE COVERING CODES IN THE Q-ARY HYPERCUBE

We investigate a generalized version of the Pathological Liar Game of Ellis and Yan. In our version, there are nonnegative integer parameters M,n, e, a, q with 0 < a < q. The player (Carole) is equipped with M messages, each with an integral error count. Carole performs the following procedure n times in sequence: she numbers the messages in increasing order of error count, divides them into consecutive contiguous size-q blocks, and selects a size-a subset of each block. The messages she does not select have their error count increased by 1. Carole loses iff, after this process is complete, there is at least one message with error count ≤ e. We allow n to approach infinity and suppose e = bfnc for a fixed f ∈ (0, 1). We establish an upper bound on the minimum M for which Carole always loses, as a function of n, by extending a technique of Cooper and Ellis; we develop a simple process that approximates the course of the game for any choice by Carole, and bound the difference between this approximating process and any state of the game achievable by Carole. Along the way we generalize several of the results of Cooper and Ellis in ways that suggest future application to similar problems in adaptive coding theory.