Two Approaches to the Construction of Deletion Correcting Codes: Weight Partitioning and Optimal Colorings

TL;DR

This paper explores two graph-theoretic methods for constructing binary deletion correcting codes, focusing on weight partitioning and optimal colorings, and demonstrates their near-optimality through theoretical bounds and colorings.

Contribution

It introduces a new approach using constant Hamming weight codes and provides optimal colorings for single deletion cases, advancing the understanding of deletion code construction.

Findings

Union of constant weight codes yields near-optimal size bounds.

Optimal colorings of weight-induced subgraphs are asymptotically optimal.

VT codes are shown to be optimal in a coloring sense.

Abstract

We consider the problem of constructing deletion correcting codes over a binary alphabet and take a graph theoretic view. An $n$-bit $s$-deletion correcting code is an independent set in a particular graph. We propose constructing such a code by taking the union of many constant Hamming weight codes. This results in codes that have additional structure. Searching for codes in constant Hamming weight induced subgraphs is computationally easier than searching the original graph. We prove a lower bound on size of a codebook constructed this way for any number of deletions and show that it is only a small factor below the corresponding lower bound on unrestricted codes. In the single deletion case, we find optimal colorings of the constant Hamming weight induced subgraphs. We show that the resulting code is asymptotically optimal. We discuss the relationship between codes and colorings and observe that the VT codes are optimal in a coloring sense. We prove a new lower bound on the chromatic number of the deletion channel graphs. Colorings of the deletion channel graphs that match this bound do not necessarily produce asymptotically optimal codes.