Generic Reed-Solomon codes achieve list-decoding capacity

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-\(ℓ\) MDS code, denoted by \(MDS(ℓ)\), has the property that any \(ℓ\) 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 \(MDS(ℓ)\) for all \(ℓ\), 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 \(1-R-\epsilon\) with list size at most (\(\frac{1-R-\epsilon )/}{\epsilon }\), resolving a conjecture of Shangguan and Tamo.