Robust Gray Codes Approaching the Optimal Rate

Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code G so that, given a noisy version of the encoding G(j) of an integer j, one can recover j that is close to j (with high probability over the noise). Such codes have found applications in differential privacy. In this work, we present near-optimal constructions of robust Gray codes. In more detail, we construct a Gray code G of rate 1-H2(p)-ϵ that is efficiently encodable, and that is robust in the following sense. Supposed that G(j) is passed through the binary symmetric channel BSCp with cross-over probability p, to obtain x. We present an efficient decoding algorithm that, given x, returns an estimate j so that |j -j | is small with high probability.