Improved Constructions of Linear Codes for Insertions and Deletions

2025-09-30

AI Summary

This work addresses the problem of approaching the semi-Singleton bound for linear error-correcting codes under adversarial insertion-deletion (indel) errors. Existing linear indel codes suffer from low rate, non-explicit constructions, and inefficient encoding/decoding. To overcome these bottlenecks, we propose an explicit algebraic construction based on subfield linearity: codes that are linear over $mathbb{F}_{q^2}$ and simultaneously linear over the subfield $mathbb{F}_q$, thereby achieving both high rate and efficient encoding/decoding. Our first construction attains rate $1/2 - delta - varepsilon$ while efficiently correcting a $delta$-fraction of indel errors—marking the first explicit linear code with such rate–correction trade-off. We further construct fully $mathbb{F}_q$-linear codes achieving rate $1/2 - 2sqrt{delta} - varepsilon$, significantly improving the rate–error-correction trade-off for linear indel codes. Moreover, our results generalize the semi-Singleton bound to broader classes of linear code structures.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over (mathbb{F}_{q^2}), linear over (mathbb{F}_q), with rate (1/2 - δ- varepsilon) that can efficiently correct a (δ)-fraction of indel errors, where (q = O(varepsilon^{-4})). Additionally, we construct fully linear codes over (mathbb{F}_q) with rate (1/2 - 2sqrtδ - varepsilon) that can also efficiently correct (δ)-fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code (C subseteq mathbb{F}^n) linear over (mathbb{E} subset mathbb{F}) a subfield of $mathbb{F}$, such that (C) has the ability to correct (δ)-fraction of indels, the rate is bounded by $(1-δ)/2$.

Problem

• Constructing linear codes correcting adversarial insertion-deletion errors efficiently
• Achieving near half-Singleton bound rates for linear codes over finite fields
• Developing explicit and fully linear codes with efficient indel error correction

Innovation

• Linear codes over F_q^2 correct indel errors efficiently
• Fully linear codes over F_q achieve near half-Singleton bound
• Generalized half-Singleton bound for subfield linear codes