The $b$-adic diaphony of digital $(\mathbf T, s)$-sequences

The $b$-adic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this article we give an upper bound on the $b$-adic diaphony of digital $(\mathbf T, s)$-sequences over $\mathbb Z_b$. And we derive a condition on the quality function $\mathbf T$ such that the $b$-adic diaphony of the digital $(\mathbf T, s)$-sequence over $\mathbb Z_b$ is of order $\mathcal O((\log N)^{s/2}N^{-1})$. We also give a metrical result; for $\mu_s$-almost all generators of a digital $(\mathbf T,s )$-sequence over $\mathbb Z_b$ the $b$-adic diaphony of the sequence is of order$\mathcal O((\log \log N)^2 (\logN)^{3s/2}N^{-1})$.