Clique-Based Deletion-Correcting Codes via Penalty-Guided Clique Search

TL;DR

This paper introduces a novel heuristic for constructing larger deletion-correcting codes by formulating the problem as a maximum clique search and applying a penalty-guided heuristic, outperforming existing methods and matching known optimal sizes.

Contribution

It proposes a penalty-guided clique search heuristic for constructing deletion-correcting codes, improving code sizes over existing graph-based heuristics and classical constructions.

Findings

PGCS yields larger codebooks than existing heuristics.

Codebooks match or exceed known optimal sizes for certain parameters.

Optimized LCS-based decoder reduces decoding complexity significantly.

Abstract

We study the construction of $d$-deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common subsequence (LCS) distance guarantees correction of up to $d$ deletions. A valid codebook corresponds to a clique in the resulting graph, and finding the largest codebook is equivalent to identifying a maximum clique. While MCP-based formulations for deletion-correcting codes have previously been explored, we demonstrate that applying Penalty-Guided Clique Search (PGCS), a lightweight stochastic clique-search heuristic inspired by Dynamic Local Search (DLS), consistently yields larger codebooks than existing graph-based heuristics, including minimum-degree and coloring methods, for block lengths $n = 8,9,\dots,14$ and deletion parameters $d = 1,2,3$. In several finite-length regimes, the resulting codebooks match known optimal sizes and outperform classical constructions such as Helberg codes. For decoding under segmented reception, where codeword boundaries are known, we propose an optimized LCS-based decoder that exploits symbol-count filtering and early termination to substantially reduce the number of LCS evaluations while preserving exact decoding guarantees. These optimizations lead to significantly lower average-case decoding complexity than the baseline $O(|C| n^2)$ approach.