A Two Prover One Round Game with Strong Soundness

We show that for any fixed prime $q \geq 5$ and constant $\zeta >, 0$, it is NP-hard to distinguish whether a two prove one round game with $q^6$ answers has value at least $1-\zeta$ or at most $\frac{4}{q}$. The result is obtained by combining two techniques: (i) An Inner PCP based on the {\it point versus subspace} test for linear functions. The testis analyzed Fourier analytically. (ii) The Outer/Inner PCP composition that relies on a certain {\it sub-code covering} property for Hadamard codes. This is a new and essentially black-box method to translate a {\it codeword test}for Hadamard codes to a {\it consistency test}, leading to a full PCP construction. As an application, we show that unless NP has quasi-polynomial time deterministic algorithms, the Quadratic Programming Problem is in approximable within factor $(\log n)^{1/6 - o(1)}$.