Improved Construction of Robust Gray Code (2401.15291v1)
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 $\Omega(R)$, 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 $R/2$.
- D. R. Lolck and R. Pagh, “Shannon meets gray: Noise-robust, low-sensitivity codes with applications in differential privacy,” in Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2024, pp. 1050–1066.
- M. Aumüller, C. J. Lebeda, and R. Pagh, “Differentially private sparse vectors with low error, optimal space, and fast access,” in Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security, 2021, pp. 1223–1236.
- J. Acharya, Y. Liu, and Z. Sun, “Discrete distribution estimation under user-level local differential privacy,” in International Conference on Artificial Intelligence and Statistics. PMLR, 2023, pp. 8561–8585.
- J. Acharya, C. Canonne, Y. Liu, Z. Sun, and H. Tyagi, “Distributed estimation with multiple samples per user: Sharp rates and phase transition,” Advances in neural information processing systems, vol. 34, pp. 18 920–18 931, 2021.
- L. Xiao, X.-G. Xia, and Y.-P. Wang, “Exact and robust reconstructions of integer vectors based on multidimensional chinese remainder theorem (md-crt),” IEEE Transactions on Signal Processing, vol. 68, pp. 5349–5364, 2020.
- W. Wang and X.-G. Xia, “A closed-form robust chinese remainder theorem and its performance analysis,” IEEE Transactions on Signal Processing, vol. 58, no. 11, pp. 5655–5666, 2010.
- G. Reeves and H. D. Pfister, “Reed–muller codes on bms channels achieve vanishing bit-error probability for all rates below capacity,” IEEE Transactions on Information Theory, 2023.
- E. Arikan, “A performance comparison of polar codes and reed-muller codes,” IEEE Communications Letters, vol. 12, no. 6, pp. 447–449, 2008.
- V. Guruswami and P. Xia, “Polar codes: Speed of polarization and polynomial gap to capacity,” IEEE Transactions on Information Theory, vol. 61, no. 1, pp. 3–16, 2014.
- M. Mondelli, S. H. Hassani, and R. L. Urbanke, “Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors,” IEEE Transactions on Information Theory, vol. 62, no. 12, pp. 6698–6712, 2016.
- H.-P. Wang, T.-C. Lin, A. Vardy, and R. Gabrys, “Sub-4.7 scaling exponent of polar codes,” IEEE Transactions on Information Theory, 2023.
- F. Gray, “Pulse code communication,” Mar. 17 1953, uS Patent 2,632,058.
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