Short PCPPs verifiable in polylogarithmic time with O(1) queries

In this paper we show for every pair language $L\subseteq \{0,1\}^*\times\{0,1\}^*$ in ${\ensuremath{\mathsf{NTIME}}}(T)$ for some non-decreasing function $T:{{\mathbb Z}}^+\rightarrow {{\mathbb Z}}^+$ there is a ${\ensuremath{\mathsf{PCPP}}}$-verifier such that the following holds. In time poly (|x|,log|y|,logT(|x| + |y|)) it decides the membership of a purported word (x,y) by reading the explicit input x entirely and querying the implicit input y and the auxiliary proof of length T(|x| + |y|)·poly log T(|x| + |y|) in a constant number of positions.