  We study quasi-Monte Carlo integration for twice differentiable functions
defined over a triangle. We provide an explicit construction of infinite
sequences of points including one by Basu and Owen (2015) as a special case,
which achieves the integration error of order $N^{-1}(\log N)^3$ for any $N\geq
2$. Since a lower bound of order $N^{-1}$ on the integration error holds for
any linear quadrature rule, the upper bound we obtain is best possible apart
from the $\log N$ factor. The major ingredient in our proof of the upper bound
is the dyadic Walsh analysis of twice differentiable functions over a triangle
under a suitable recursive partitioning.
