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TR07-026 | 21st November 2006 00:00
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#### Sub-Constant Error Probabilistically Checkable Proof of Almost Linear Size

**Abstract:**
We show a construction of a PCP with both sub-constant error and

almost-linear size. Specifically, for some constant alpha in (0,1),

we construct a PCP verifier for checking satisfiability of

Boolean formulas that on input of size n uses log n + O((log

n)^{1-alpha}) random bits to query a constant number of places

in a proof of size n 2^{O((log n)^{1-alpha})} over

symbols consisting of O((log n)^{1-alpha}) bits and achieves

error 2^{-Omega((log n)^{alpha})}.

The construction is by a new randomness-efficient version of

the aggregation through curves technique. Its main

ingredients are a recent low degree test with both

sub-constant error and almost-linear size and a new

method for constructing a short list of balanced curves.