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.

翻译：鲁棒格雷码由（Lolck与Pagh，SODA 2024）提出。非正式地说，鲁棒格雷码是一种（二进制）格雷码$\mathcal{G}$，使得在给定整数$j$的编码$\mathcal{G}(j)$的噪声版本后，能够以高概率（在噪声分布上）恢复出接近$j$的估计值$\hat{j}$。此类编码已在差分隐私领域得到应用。本工作中，我们提出了接近最优的鲁棒格雷码构造。具体而言，我们构造了一种编码效率为$1 - H_2(p) - \varepsilon$的格雷码$\mathcal{G}$，该编码具有高效可编码性，并满足以下鲁棒性：假设$\mathcal{G}(j)$通过交叉概率为$p$的二进制对称信道$\text{BSC}_p$后得到$x$。我们提出一种高效解码算法，在给定$x$时返回估计值$\hat{j}$，使得$|j - \hat{j}|$以高概率保持较小值。