Improved Construction of Robust Gray Code
Dorsa Fathollahi, Mary Wootters
TL;DR
This work advances the construction of robust Gray codes by showing how a linear binary code with rate $R$ can be transformed into a robust Gray code whose rate approaches $R/2$, while preserving robustness guarantees similar to prior work. The authors introduce a three-stage pipeline: order the base code $\mathcal{C}$, build an intermediate code $\mathcal{W}$ via parity-augmented copies, and interpolate between $\mathcal{W}$ codewords to produce the final Gray code $\mathcal{G}$ with encoding $\mathrm{Enc}_{\mathcal{G}}$. The key novelty lies in how $\mathcal{W}$ is constructed and how the crossover position is detected, enabling a decoding strategy that locates the correct segment and decodes using the original code $\mathcal{C}$, with a rate approaching $R/2$ and poly-time encoding/decoding. The resulting robust Gray codes are well-suited for applications in differential privacy and related areas, and can be instantiated with standard codes like Reed-Muller or polar codes to achieve efficient, noise-robust encoding with near-half-rate efficiency. Overall, the paper achieves a practical improvement in the trade-off between robustness and rate for Gray codes in binary settings.
Abstract
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 $Ω(R)$, where the constant in the $Ω(\cdot)$ 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 $R/2$.