Capacity-Achieving Gray Codes

TL;DR

This paper introduces a novel implementation of Gray codes that achieves the capacity of the binary symmetric channel with positive error exponents, enhancing privacy-preserving integer encoding.

Contribution

It proposes a capacity-achieving Gray code design with integrated error correction, improving privacy-preserving data release methods.

Findings

Achieves BSC capacity with positive error exponents

Enhances differential privacy mechanisms for integer data

Introduces a new coding scheme inspired by software backup processes

Abstract

To ensure differential privacy, one can reveal an integer fuzzily in two ways: (a) add some Laplace noise to the integer, or (b) encode the integer as a binary string and add iid BSC noise. The former is simple and natural while the latter is flexible and affordable, especially when one wants to reveal a sparse vector of integers. In this paper, we propose an implementation of (b) that achieves the capacity of the BSC with positive error exponents. Our implementation adds error-correcting functionality to Gray codes by mimicking how software updates back up the files that are getting updated ("coded Gray code"). In contrast, the old implementation of (b) interpolates between codewords of a black-box error-correcting code ("Grayed code").