Hitting Sets for Low-Degree Polynomials with Optimal Density

We give a length-efficient puncturing of Reed-Muller codes which preserves its distance properties. Formally, for the Reed-Muller code encoding n-variate degree-d polynomials over Fq with q ≳ d/δ, we present an explicit (multi)-set S ⊆ Fqn of size N=poly(nd/δ) such that every nonzero polynomial vanishes on at most delta N points in S. Equivalently, we give an explicit hitting set generator (HSG) for degree-d polynomials of seed length log N = O(d log n + log (1/δ)) with "density" 1-δ (meaning every nonzero polynomial is nonzero with probability at least 1-δ on the output of the HSG). The seed length is optimal up to constant factors, as is the required field size Omega(d/delta). Plugging our HSG into a construction of Bogdanov (STOC'05) gives explicit pseudorandom generators for n-variate degree-d polynomials with error eps and seed length O(d4 log n + log (1/ε)) whenever the field size satisfies q gtrsim d6/ε2. Our approach involves concatenating previously known HSGs over large fields with multiplication friendly codes based on algebraic curves. This allows us to bring down the field size to the optimal bounds. Such multiplication friendly codes, which were first introduced to study the bilinear complexity of multiplication in extension fields, have since found other applications, and in this work we give a further use of this notion in algebraic pseudorandomness.