Generic Reed-Solomon Codes Achieve List-decoding Capacity
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| Format: | Preprint |
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2022
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| author | Brakensiek, Joshua Gopi, Sivakanth Makam, Visu |
| author_facet | Brakensiek, Joshua Gopi, Sivakanth Makam, Visu |
| contents | In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-$\ell$ MDS code, denoted by $\operatorname{MDS}(\ell)$, has the property that any $\ell$ subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth defined a different notion of higher order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher order MDS codes are (nearly) equivalent.
We also show that generic Reed-Solomon codes are $\operatorname{MDS}(\ell)$ for all $\ell$, relying crucially on the GM-MDS theorem which shows that generator matrices of generic Reed-Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed-Solomon codes achieve list decoding capacity. More concretely, we show that, with high probability, a random Reed-Solomon code of rate $R$ over an exponentially large field is list decodable from radius $1-R-ε$ with list size at most $\frac{1-R-ε}ε$, resolving a conjecture of Shangguan and Tamo. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_05256 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generic Reed-Solomon Codes Achieve List-decoding Capacity Brakensiek, Joshua Gopi, Sivakanth Makam, Visu Information Theory Computational Complexity Combinatorics In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-$\ell$ MDS code, denoted by $\operatorname{MDS}(\ell)$, has the property that any $\ell$ subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth defined a different notion of higher order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher order MDS codes are (nearly) equivalent. We also show that generic Reed-Solomon codes are $\operatorname{MDS}(\ell)$ for all $\ell$, relying crucially on the GM-MDS theorem which shows that generator matrices of generic Reed-Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed-Solomon codes achieve list decoding capacity. More concretely, we show that, with high probability, a random Reed-Solomon code of rate $R$ over an exponentially large field is list decodable from radius $1-R-ε$ with list size at most $\frac{1-R-ε}ε$, resolving a conjecture of Shangguan and Tamo. |
| title | Generic Reed-Solomon Codes Achieve List-decoding Capacity |
| topic | Information Theory Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2206.05256 |