Ghasemi-Kopparty Problem on Sparse S-Decoding Polynomials

Can $S$-decoding polynomials modulo a product of $k$ primes be built with only $k+1$ nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any constant $s$ there is an $s$-server private information retrieval protocol with communication $\exp(O((\log n)^{1/s}(\log\log n)^{1-1/s}))$ on an $n$-bit database, where previous constructions at that communication needed $2^{O(s)}$ servers.