The improvements in the constants has many times been obtained by extracting some important property from a previous protocol, using that protocol as a black box and then adding some conceptually new construction. This is more or less what we do in the current paper. … The long code is universal in that it contains every other binary code as a sub-code. Thus it never hurts to have this code available, but it is still surprising that it is beneficial to have such a wasteful code.
We saw in Chapter 11 that the PCP Theorem implies that computing approximate solutions to many optimization problems is NP-hard. This chapter gives a complete proof of the PCP Theorem. In Chapter 11 we also mentioned that the PCP Theorem does not suffice for proving several other similar results, for which we need stronger (or simply different) “PCP Theorems.” In this chapter we survey some such results and their proofs. The two main results are Raz's Parallel Repetition Theorem (see Section 22.3) and Håstad's Three-Bit PCP Theorem (Theorem 22.16). Raz's theorem leads to strong hardness results for the 2CSP problem over large alphabets. Håstad's theorem shows that certificates for NP languages can be probabilistically checked by examining only three bits in them.