A two prover one round game with strong soundness

We show that for any fixed prime q ≥ 5 and constant ζ > 0, it is NP-hard to distinguish whether a two prove one round game with q ⁶ answers has value at least 1-ζ or at most 4/q. The result is obtained by combining two techniques: (i) An Inner PCP based on the point versus subspace test for linear functions. The testis analyzed Fourier analytically. (ii) The Outer/Inner PCP composition that relies on a certain sub-code covering property for Hadamard codes. This is a new and essentially black-box method to translate a codeword test for Hadamard codes to a 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)}.