The content presents a new result that improves upon previous work on the alphabet-soundness tradeoff for 2-query PCPs. The key contributions are:
The main technical result (Theorem 1.3) shows that for all ε, δ > 0, it is NP-hard to distinguish whether a 2-Prover-1-Round game with alphabet size q has value at least 1-δ or at most 1/q^(1-ε). This establishes a nearly optimal tradeoff between the alphabet size and soundness error of PCPs.
This improved tradeoff has several applications, including:
The technical approach involves composing an "inner PCP" based on the Grassmann graph with an "outer PCP" using smooth parallel repetition. The analysis requires new techniques in low-degree testing and list decoding over the Grassmann graph.
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