But I’m glad I thought about this, because the uncertainty formulation is a really nice question! I’m going to think about this more.

]]>Notation: Set . So . We want a polynomial in variables which is at the origin and is supported on the Hamming ball of radius . In order to get a nontrivial bound, we need . Asymptotically, this means we need .

We may as well assume that uses only square-free monomials. The coefficient of the monomial in is where we sum over with for .

Now, it seems to me that the easiest way to get supported on the Hamming ball is that is a function of the Hamming weight of ; say . So the coefficient of is . Call this . We see that is an integer-valued polynomial of degree with for . Also, since , the constant term of is nonzero and .

But induction on shows that, if is a nonzero integer valued polynomial with consecutive zeroes, then . (Base case is clear, then replace by for the induction.) So and we get . Set and . So and we want . Also, obviously, and .

But is convex down, so the minimum of must occur at one of the corners of the triangle , and the values at the corners are , and . We lose.

]]>