4.1 Walsh-hadamard Code (recap from Last Lecture)

In today’s lecture, we will construct exponential sized PCPs for NP. More formally, we will show that the NP-Complete problem, Circuit-Satisfiability (Circuit-SAT) is in PCP1,1−δ[O(n), O(1)] for some 0 < δ < 1. Recall that the PCP Theorem states that Circuit-SAT ∈ PCP1,1−δ[O(log n), O(1)]. However, as we will see in later lectures, the exponential sized PCPs for Circuit-SAT will be used in proving the PCP Theorem. Actually, we will prove a slightly stronger result than Circuit-SAT ∈ PCP1,1−δ[O(log n), O(1)]. We will actually construct a PCP of Proximity for a related problem, Circuit-Value (Circuit-VAL).