{"id":851653,"date":"2022-06-12T19:09:13","date_gmt":"2022-06-13T02:09:13","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/"},"modified":"2023-02-06T14:18:28","modified_gmt":"2023-02-06T22:18:28","slug":"generic-reed-solomon-codes-achieve-list-decoding-capacity","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/generic-reed-solomon-codes-achieve-list-decoding-capacity\/","title":{"rendered":"Generic Reed-Solomon codes achieve list-decoding capacity"},"content":{"rendered":"

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-\u2113<\/span><\/span><\/span><\/span>\u00a0MDS code, denoted by\u00a0MDS<\/span><\/span>(<\/span>\u2113<\/span>)<\/span><\/span><\/span><\/span>, has the property that any\u00a0\u2113<\/span><\/span><\/span><\/span>\u00a0subspaces 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.
\nWe also show that generic Reed-Solomon codes are\u00a0MDS<\/span><\/span>(<\/span>\u2113<\/span>)<\/span><\/span><\/span><\/span>\u00a0for all\u00a0\u2113<\/span><\/span><\/span><\/span>, 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\u00a0R<\/span><\/span><\/span><\/span>\u00a0over an exponentially large field is list decodable from radius\u00a01<\/span>\u2212<\/span>R<\/span>\u2212<\/span>\u03f5<\/span><\/span><\/span><\/span> with list size at most (1<\/span>\u2212<\/span>R<\/span>\u2212<\/span>\u03f5)\/<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>, resolving a conjecture of Shangguan and Tamo.<\/p>\n","protected":false},"excerpt":{"rendered":"

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-\u2113\u00a0MDS code, denoted by\u00a0MDS(\u2113), has the property that any\u00a0\u2113\u00a0subspaces 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 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