--Existence: there exists a received word w_N : \mathbb{F}_N \to \mathbb{F}_N that agrees with a super-polynomial number of distinct degree K polynomials on \approx N^{\sqrt \delta} points each;

--Explicit: there exists a polynomial time constructible received word w'_N : \mathbb{F}_N \to \mathbb{F}_N that agrees with a super-polynomial number of distinct degree K polynomials, on \approx 2^{\sqrt {\log N} } K points each.

In both cases, our results improve upon the previous state of the art, which was \approx N^{\delta}/\delta for the existence case [JH01], and \approx 2N^{\delta} for the explicit one [GR05b]. Furthermore, for \delta close to 1 our bound approaches the Guruswami-Sudan bound (which is \sqrt {NK}) and implies limitations on extending their efficient RS list decoding algorithm to larger decoding radius.

Our proof method is surprisingly simple. We work with polynomials that vanish on subspaces of an extension field viewed as a vector space over the base field. These subspace polynomials are a subclass of linearized polynomials that were first studied by Ore [Ore33, Ore34] in the 1930s, and later by coding theorists. For us their main attraction is their sparsity and abundance of roots, virtues that recently won them pivotal roles in probabilistically checkable proofs of proximity [BSGH+04, BSS05] and sub-linear proof verification [BSGH+05].