A robust Gray code, formally introduced by (Lolck and Pagh, SODA 2024), is a Gray code that additionally has the property that, given a noisy version of the encoding of an integer $j$, it is possible to reconstruct $\hat{j}$ so that $|j - \hat{j}|$ is small with high probability. That work presented a transformation that transforms a binary code $C$ of rate $R$ to a robust Gray code with rate $\Omega(R)$, where the constant in the $\Omega(\cdot)$ can be at most $1/4$. We improve upon their construction by presenting a transformation from a (linear) binary code $C$ to a robust Gray code with similar robustness guarantees, but with rate that can approach $R/2$.

翻译：鲁棒格雷码由(Lolck和Pagh, SODA 2024)正式提出，是一种具有额外性质的格雷码：给定整数$j$的编码的含噪版本，能够以高概率重构出$\hat{j}$，使得$|j - \hat{j}$较小。该工作提出了一种变换，将速率为$R$的二进制码$C$转换为速率为$\Omega(R)$的鲁棒格雷码，其中$\Omega(\cdot)$中的常数至多为$1/4$。我们改进了他们的构造，提出了一种从（线性）二进制码$C$到鲁棒格雷码的变换，具有相似的鲁棒性保证，但其速率可接近$R/2$。