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Robust Gray Codes Approaching the Optimal Rate

Roni Con, Dorsa Fathollahi, Ryan Gabrys, Mary Wootters, Eitan Yaakobi

TL;DR

This work presents an efficient decoding algorithm that returns an estimate of j so that, given x, one can recover an estimate of j that is close to j (with high probability over the noise).

Abstract

Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code $\mathcal{G}$ so that, given a noisy version of the encoding $\mathcal{G}(j)$ of an integer $j$, one can recover $\hat{j}$ that is close to $j$ (with high probability over the noise). Such codes have found applications in differential privacy. In this work, we present near-optimal constructions of robust Gray codes. In more detail, we construct a Gray code $\mathcal{G}$ of rate $1 - H_2(p) - \varepsilon$ that is efficiently encodable, and that is robust in the following sense. Supposed that $\mathcal{G}(j)$ is passed through the binary symmetric channel $\text{BSC}_p$ with cross-over probability $p$, to obtain $x$. We present an efficient decoding algorithm that, given $x$, returns an estimate $\hat{j}$ so that $|j - \hat{j}|$ is small with high probability.

Robust Gray Codes Approaching the Optimal Rate

TL;DR

This work presents an efficient decoding algorithm that returns an estimate of j so that, given x, one can recover an estimate of j that is close to j (with high probability over the noise).

Abstract

Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code so that, given a noisy version of the encoding of an integer , one can recover that is close to (with high probability over the noise). Such codes have found applications in differential privacy. In this work, we present near-optimal constructions of robust Gray codes. In more detail, we construct a Gray code of rate that is efficiently encodable, and that is robust in the following sense. Supposed that is passed through the binary symmetric channel with cross-over probability , to obtain . We present an efficient decoding algorithm that, given , returns an estimate so that is small with high probability.
Paper Structure (27 sections, 10 theorems, 65 equations, 2 figures, 4 algorithms)

This paper contains 27 sections, 10 theorems, 65 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Independent Work.
  4. Technical Overview
  5. "Interpolating" between codewords of an intermediate code.
  6. Defining the intermediate code.
  7. Decoding the resulting robust Gray code.
  8. Definitions and Construction
  9. Notation and useful definitions
  10. Case 1: $i < 2^{k-1}$.
  11. Case 2: $i \geq 2^{k-1}$.
  12. Base code
  13. Ingredients.
  14. Ordering the base code.
  15. Intermediate Code
  16. ...and 12 more sections

Key Result

Theorem 1

Fix constants $p \in (0,1/2)$ and a sufficiently small $\varepsilon > 0$. Fix a constant $\mathcal{R} \in (0,1)$. Let $d$ be sufficiently large, in terms of these constants. Then there is an $n' = \Theta(\log d)$ so that the following holds. Suppose that there exists a binary linear $[n',k']_2$ code

Figures (2)

  • Figure 1: The notation used to break up vectors $x \in \{0,1\}^d$ into chunks (top), and the distinction between chunks and full chunks when $x$ happens to be a codeword $g_j$ (bottom). Notice that for $g_j$, if $j \in [r_i, r_{i+1})$ then we have, e.g., $s_m = b_i^{B}$ and $\tilde{c}_m = c_i[m]$ or $c_{i+1}[m]$, whenever the corresponding chunks are full chunks.
  • Figure 2: Two cases for where $\hat{\ell}$ can land. As one case see in Case 2, it can be the case that $\ell$ is in the end of the transmitted codeword whereas $\hat{\ell}$, our estimate of $\ell$, is in the beginning.

Theorems & Definitions (43)

  • Theorem 1
  • Remark 1: The running time of $\mathrm{Dec}_{\mathcal{C}_{\text{in}}}$
  • Corollary 1
  • proof
  • Lemma 1: Multiplicative Chernoff bound; see, e.g., mitzenmacher2017probability
  • Lemma 2: Hoeffding's Inequality; see, e.g., mitzenmacher2017probability
  • Definition 1: Binary Reflected Code, gray
  • Definition 2
  • Example 1
  • proof
  • ...and 33 more