Query efficient PCPs with perfect completeness

For every integer k>1, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k+k² bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with a certain maximum probability. In particular, the verifier achieves optimal amortized query complexity of 1+δ for arbitrarily small constant δ>0. Such a characterization was already proved by A. Samorodnitsky and L. Trevisan (2000), but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier, we can decrease the number of query bits to 2k+k², the same number obtained by Samorodnitsky and Trevisan. Finally, we extend some of the results to larger domains.