Robust Gray Codes Approaching the Optimal Rate

Abstract

Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code $\mathcal {G}$ so that, given a noisy version of the encoding $\mathcal {G}(j)$ of an integer j, one can recover $\hat {j}$ 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 $\mathcal {G}$ of rate $1 - H_{2}(p) - \varepsilon $ that is efficiently encodable, and that is robust in the following sense. Supposed that $\mathcal {G}(j)$ is passed through the binary symmetric channel ${\text {BSC}}_{p}$ with cross-over probability p, to obtain x. We present an efficient decoding algorithm that, given x, returns an estimate $\hat {j}$ so that $| j - \hat {j}|$ is small with high probability.