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The Hamming oracle returns the Hamming distance between an unknown binary$n$-vector $x$ and a binary query $n$-vector y. The objective is to determine$x$ uniquely using a sequence of $m$ queries. What are the minimum number ofqueries required in the worst case? We consider the query ratio $m/n$ to be ourfigure of merit and derive upper bounds on the query ratio by explicitlyconstructing $(m,n)$ query matrices. We show that our recursive and algebraicconstruction results in query ratios arbitrarily close to zero. Ourconstruction is based on codes of constant weight. A decoding algorithm forrecovering the unknown binary vector is also described.

Query Matrices for Retrieving Binary Vectors Based on the Hamming Distance Oracle