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\u00a0\\(MDS(\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 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\u00a0\\(MDS(\u2113)\\)\u00a0for all\u00a0\\(\u2113\\), 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\u00a0\\(R\\)\u00a0over an exponentially large field is list decodable from radius\u00a0\\(1-R-\\epsilon\\) with list size at most (\\(\\frac{1-R-\\epsilon )\/}{\\epsilon }\\), 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-\u00a0MDS code, denoted by\u00a0, has the property that any\u00a0\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 […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Joshua Brakensiek","user_id":0},{"type":"user_nicename","value":"Sivakanth Gopi","user_id":"37830"},{"type":"text","value":"Visu Makam","user_id":0}],"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC 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