Improved Construction of Robust Gray Codes

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 $\vert j-\hat{j}\vert$ is small with high probability. That work presented a transformation that transforms a binary code $\mathcal{C}$ 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 $\mathcal{C}$ to a robust Gray code with similar robustness guarantees, but with rate that can approach $R/2$. A full version of this paper can be found in [1].