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arXiv · 2504.03090

Optimal Erasure Codes and Codes on Graphs

Yeyuan Chen·Mahdi Cheraghchi·Nikhil Shagrithaya
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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.

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Yeyuan Chen, Mahdi Cheraghchi, Nikhil Shagrithaya. 2025-04-03. Optimal Erasure Codes and Codes on Graphs. https://arxiv.org/abs/2504.03090

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