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

TL;DR

This paper develops explicit constructions of query matrices for efficiently identifying an unknown binary vector using Hamming distance queries, achieving near-zero query ratios with algebraic and recursive methods.

Contribution

It introduces novel algebraic and recursive constructions of query matrices that minimize the number of queries needed for exact recovery.

Findings

Query ratios can be made arbitrarily close to zero.

Explicit constructions based on constant weight codes.

A decoding algorithm for vector recovery is provided.

Abstract

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 of queries required in the worst case? We consider the query ratio $m/n$ to be our figure of merit and derive upper bounds on the query ratio by explicitly constructing $(m,n)$ query matrices. We show that our recursive and algebraic construction results in query ratios arbitrarily close to zero. Our construction is based on codes of constant weight. A decoding algorithm for recovering the unknown binary vector is also described.