Improved Constructions of Skew-Tolerant Gray Codes

TL;DR

This paper introduces new skew-tolerant Gray code constructions with exponential improvements, efficient encoding/decoding algorithms, and extensions to non-binary alphabets, advancing the state of the art in Gray code design.

Contribution

It provides the first asymptotically non-vanishing skew-tolerant Gray codes, along with linear-time algorithms and generalizations to non-binary alphabets.

Findings

Exponential improvement over previous constructions

Linear-time encoding and decoding algorithms

Complete m-ary skew-tolerant Gray codes for all m ≥ 3

Abstract

We study skew-tolerant Gray codes, which are Gray codes in which changes in consecutive codewords occur in adjacent positions. We present the first construction of asymptotically non-vanishing skew-tolerant Gray codes, offering an exponential improvement over the known construction. We also provide linear-time encoding and decoding algorithms for our codes. Finally, we extend the definition to non-binary alphabets, and provide constructions of complete $m$-ary skew-tolerant Gray codes for every base $m\geq 3$.