Robust Gray Codes Approaching the Optimal Rate

TL;DR

This paper introduces near-optimal, efficiently encodable robust Gray codes that can recover original indices from noisy encodings with high probability, advancing applications in differential privacy.

Contribution

The authors construct Gray codes with near-optimal rate that are robust against noise and have efficient encoding and decoding algorithms, improving upon prior work.

Findings

Constructed Gray codes with rate close to 1 - H_2(p) - ε.

Developed efficient encoding and decoding algorithms for noisy environments.

Achieved high-probability accurate index recovery from noisy codewords.

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.