List decoding and Higher Order MDS codes

A [n,k] code C is a k-dimensional subspace in F_q^n. C is said to be MDS (maximal distance separable) if its hamming weight is n-k+1 which is the largest possible by the singleton bound. It is also equivalent to saying that every k minor of the generator matrix of C is invertible. This can be rewritten as saying that any two subspaces formed by the columns of the generator matrix of C intersect as minimally as possible.

Brakensiek, Gopi, and Makam generalize this notion to introduce MDS(l) codes where we ask that any l subspace formed by the columns of the generator matrix of C intersect as minimally as possible. In a series of works, they show their new notion is equivalent to an alternative notion of higher-order MDS codes introduced by Roth related to list decoding. In particular, the parity check matrix of code being MDS(l) is equivalent to having (average-case) combinatorial list decoding guarantees. They also show that being MDS(l) is equivalent to the generator matrix being able to achieve generic zero patterns. Using the GM-MDS theorem which says that any Reed-Solomon codes achieve any generic zero pattern this implies that they generically achieve list decoding capacity.