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 ĵ 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 ĵ so that |j - ĵ| is small with high probability.