Optimal Erasure Codes and Codes on Graphs
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.
Cite
@article{arxiv.2504.03090,
title = {Optimal Erasure Codes and Codes on Graphs},
author = {Yeyuan Chen and Mahdi Cheraghchi and Nikhil Shagrithaya},
journal= {arXiv preprint arXiv:2504.03090},
year = {2025}
}