2022/06/10 by Brakensiek, Joshua, Gopi, Sivakanth, Makam, Visu · 6 citations
#Combinatorics (math.CO) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2206.05256
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-ε with list size at most \frac1-R-εε, resolving a conjecture of Shangguan and Tamo.