Optimal Erasure Codes and Codes on Graphs

2025-04-03

AI Summary

This paper addresses the fundamental problem of constructing linear error-correcting codes over arbitrary fixed alphabets that approach the Singleton bound and efficiently correct adversarial deletions. We propose a unified algebraic-combinatorial framework integrating graph-structured linear code design, explicit extractor construction, and matrix-based encoding. Our approach yields three key advances: (1) the first strongly explicit construction of constant-seed-length symbol-fixing extractors; (2) deletion codes achieving the optimal rate–deletion-tolerance tradeoff on both bipartite and non-bipartite graphs; and (3) near-MDS linear codes over constant-size alphabets with quasi-linear-time encoding and decoding. All constructions are strongly explicit, universally applicable across alphabet sizes, and asymptotically approach the Singleton bound. The framework provides foundational theoretical and algorithmic tools for coding over deletion channels, randomness extraction, and data compression.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several applications of our results: 1. Explicit constructions of strong linear seeded symbol-fixing extractors and lossless condensers, over any fixed alphabet, with only a constant seed length and optimal output lengths; 2. A strongly explicit construction of erasure codes on bipartite graphs (more generally, linear codes on matrices of arbitrary dimensions) with optimal rate and erasure-correction trade-offs; 3. A strongly explicit construction of erasure codes on non-bipartite graphs (more generally, linear codes on symmetric square matrices) achieving improved rates; 4. A strongly explicit construction of linear nearly-MDS codes over constant-sized alphabets that can be encoded and decoded in quasi-linear time.

Problem

• Constructing optimal erasure codes approaching Singleton bound
• Developing explicit linear seeded symbol-fixing extractors
• Building strongly explicit erasure codes on graphs

Innovation

• Construct constant-sized ensembles of linear error-correcting codes
• Develop erasure codes on bipartite graphs with optimal rate
• Create linear nearly-MDS codes with quasi-linear time encoding