Constructing Low-Redundancy Codes via Distributed Graph Coloring

TL;DR

This paper introduces a novel framework for constructing error-correcting codes using distributed graph coloring, enabling efficient encoding, decoding, and improved redundancy for various error types.

Contribution

It presents a general graph coloring-based approach for designing low-redundancy error-correcting codes with multiple new constructions and extensions.

Findings

Codes with redundancy twice the Gilbert-Varshamov bound

Hypergraph labeling for list-decodable codes

Asymptotically optimal burst error correction

Abstract

We present a general framework for constructing error-correcting codes using distributed graph coloring under the LOCAL model. Building on the correspondence between independent sets in the confusion graph and valid codes, we show that the color of a single vertex - consistent with a global proper coloring - can be computed in polynomial time using a modified version of Linial's coloring algorithm, leading to efficient encoding and decoding. Our results include: i) uniquely decodable code constructions for a constant number of errors of any type with redundancy twice the Gilbert-Varshamov bound; ii) list-decodable codes via a proposed extension of graph coloring, namely, hypergraph labeling; iii) an incremental synchronization scheme with reduced average-case communication when the edit distance is not precisely known; and iv) the first asymptotically optimal codes (up to a factor of 8) for correcting bursts of unbounded-length edits. Compared to syndrome compression, our approach is more flexible and generalizable, does not rely on a good base code, and achieves improved redundancy across a range of parameters.