Constructing Low-Redundancy Codes via Distributed Graph Coloring

TL;DR

The paper introduces a general framework that uses distributed graph coloring in the LOCAL model to construct error-correcting codes for channels with a constant number of errors, achieving polynomial-time encoding/decoding and redundancy near twice the Gilbert-Varshamov bound. It translates the confusion graph into a systematic (x, Φ(x)) code by computing a color locally, and extends the approach to list decoding via hypergraph labeling, incremental synchronization, and codes that tolerate long bursts of edits. Key contributions include a polynomial-time construction of uniquely decodable codes, a framework for list-decodable codes with constant list size, and novel synchronization protocols that reduce communication when the exact distance is unknown. The work also provides asymptotically optimal codes for bursts of unbounded-length edits and shows how syndrome compression fits within its recoloring paradigm, offering a flexible tool for designing robust edit-correcting codes across various channels and parameter regimes.

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.

Figures (3)

• Figure 1: The confusion graph for the 1-deletion channel with input $\{0,1\}^3$, along with a proper coloring. The graph coloring shown here is such that each color class corresponds to a VT code.
• Figure 2: The confusion hypergraph for the $1$-deletion channel with input $\{0,1\}^3$, along with a 2-labeling (red and blue). The edge labels, 00, 01, 10, and 11 are the output corresponding to the edge.
• Figure 3: The number of bits $B$ that Alice needs to send for synchronization based on the Oracle lower bound and three protocols (divided by $4\log n$ as a function of $p$, where the distribution of the edit distance is $\Pr(d_E(x,y)=a)=1-p,\Pr(d_E(x,y)=b)=p$.

Theorems & Definitions (70)

• Remark 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 7: Channels
• Example 1
• Definition 8: Confusion Graph and Hypergraph
• Definition 9: c.f. \ref{['rem:err-free']}