On 2-Query Codeword Testing with Near-Perfect Completeness

Abstract

A codeword tester is a highly query-efficient spot checking procedure for ascertaining, with good confidence, proximity of a given string to its closest codeword. We consider the problem of binary codeword testing using only two queries. It is known that three queries suffice for non-trivial codeword testing with perfect completeness (where codewords must be accepted with probability 1). It is known that two queries are not enough for testing with perfect completeness, whereas two queries suffice if one relaxes the requirement of perfect completeness (this is akin to the polynomial-time decidability of 2SAT and the APX-hardness of Max 2SAT, respectively).

In this work, motivated by the parallel with 2-query PCPs and the approximability of near-satisfiable instances of Max 2SAT, we investigate 2-query testing with completeness close to 1, say 1–ε for ε→0. Our result is that, for codes of constant relative distance, such testers must also have soundness 1– O(ε) (and this is tight up to constant factors in the O(ε) term). This is to be contrasted with 2-query PCPs, where assuming the Unique Games Conjecture, one can have completeness 1–ε and soundness $1-O(\sqrt{\varepsilon})$. Hence the ratio (1–s)/(1–c) can be super-constant for 2-query PCPs while it is bounded by a constant for 2-query LTCs. Our result also shows a similar limitation of 2-query PCPs of proximity, a notion introduced in [1].