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TR04-021 | 23rd March 2004 00:00

Robust PCPs of Proximity, Shorter PCPs and Applications to Coding



We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
For every $\e>0$, we present PCPs of length $\exp(\log^\e n)\cdot n$
that can be verified by making a constant number of Boolean queries.

In both cases, false assertions are rejected with
constant probability (which may be set to be arbitrarily close to~1).
The multiplicative overhead on the length of the proof,
introduced by transforming a proof into a probabilistically
checkable one, is just quasi-polylogarithmic in the first case
(of query complexity $o(\log\log n)$), and $2^{(\log n)^\epsilon}$,
for any $\epsilon > 0$, in the second case (of constant query complexity).
In contrast, previous results required at least $2^{\sqrt{\log n}}$
overhead in the length, even to get query complexity $2^{\sqrt{\log n}}$.

Our techniques include the introduction of a new variant of PCPs
that we call ``Robust PCPs of Proximity''.
These new PCPs facilitate proof composition,
which is a central ingredient in construction of PCP systems.
(A related notion and its composition properties were
discovered independently by Dinur and Reingold.)
Our main technical contribution is a construction
of a ``length-efficient'' Robust PCP of Proximity.
While the new construction uses many of the standard techniques in PCPs,
it does differ from previous constructions in fundamental ways, and in
particular does not use the ``parallelization'' step of Arora et al.
The alternative approach may be of independent interest.

We also obtain analogous quantitative results for locally testable codes.
In addition, we introduce a relaxed notion of locally decodable codes,
and present such codes mapping $k$ information bits
to codewords of length $k^{1+\e}$, for any $\e>0$.

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