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

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.

💡 Research Summary

The paper studies binary deletion‑correcting codes from a graph‑theoretic perspective. For a fixed block length n and a number s of deletions, the authors define the “deletion channel graph” Gₙ,ₛ whose vertices are all 2ⁿ binary strings and where an edge connects two strings if one can be obtained from the other by exactly s deletions. In this model, an s‑deletion‑correcting code is precisely an independent set of Gₙ,ₛ.

The first major contribution is the weight‑partitioning method. Every vertex has a Hamming weight w (the number of 1’s). By restricting attention to the induced subgraph Gₙ,ₛ^{(w)} consisting of vertices of weight w, the search space shrinks from 2ⁿ to (\binom{n}{w}). The authors show that for each weight class one can compute (or approximate) the maximum independent set size α_w, and that the union of the optimal independent sets over all weights yields a global code of size
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