We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Guruswami-Sudan bounds [Joh62, Joh63, GS99]. In particular, we show that for any ... , there exist arbitrarily large fields ... * Existence: there exists a received word ... that agrees with a super-polynomial number of distinct degree K polynomials on ... points each; * Explicit: there exists a polynomial time constructible received word ... that agrees with a super-polynomial number of distinct degree K polynomials, on ... points each. Ill both cases, our results improve upon the previous state of the art, which was , NM/6 for the existence case [JH01], and a ... for the explicit one [GR,05b]. Furthermore, for 6 close to 1 our bound approaches the Guruswami-Sudan bound (which is ... ) and rules out the possibility of extending their efficient RS list decoding algorithm to any significantly 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.
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