Three-query PCPs with perfect completeness over non-Boolean domains

We study nonBoolean PCPs that have perfect completeness and read three positions from the proof. For the case when the proof consists of values from a domain of size d for some integer constant d≥2, we construct a nonadaptive PCP with perfect completeness and soundness d^-1+d^-2+ε, for any constant ε>0, and an adaptive PCP with perfect completeness and soundness d^-1+ε, for any constant ε>0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d^-1+ε, for any constant ε>0, where four positions are read from the proof. These results match the best known constructions for the case d=2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold.