  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 $R/2$.
