On 2-Query Codeword Testing with Near-Perfect Completeness

  • Venkatesan Guruswami
Conference paper

DOI: 10.1007/11940128_28

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)
Cite this paper as:
Guruswami V. (2006) On 2-Query Codeword Testing with Near-Perfect Completeness. In: Asano T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg

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].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA

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