Locally Testable Codes Analogues to the Unique Games Conjecture Do Not Exist.

The Unique Games Conjecture (UGC ) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP veri ers: 2-query veri ers that only make unique tests. Moreover, the UGC predicts that such PCP veri ers can have almost-perfect completeness and low-soundness. The computational complexity notion of a PCP is closely related to the combinatorial notion of a Locally Testable Code (LTC ). LTCs are error-correcting codes with codeword testers that only make a constant number of queries to the tested word. All known PCP constructions use LTCs as building blocks. Furthermore, to obtain PCPs with certain properties, one uses LTCs with corresponding properties. In light of the strong connection between PCPs and LTCs, one may conjecture the existence of LTCs with properties similar to the ones required by the UGC. In this work we show that such LTCs do not exist. That is, we consider 2-query LTCs with codeword testers that only make unique tests. Roughly speaking, we show that any such LTC with relative distance close to 1, almost-perfect completeness and low-soundness, is of constant size. The result intends to point out the limitations of the current techniques for constructing PCPs. It suggests that proving the UGC may require a new approach. †Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Israel. Research supported by Binational Science Foundation (BSF) and Israel Science Foundation (ISF).