  We investigate quasi-Monte Carlo (QMC) integration of bivariate periodic
functions with dominating mixed smoothness of order one. While there exist
several QMC constructions which asymptotically yield the optimal rate of
convergence of $\mathcal{O}(N^{-1}\log(N)^{\frac{1}{2}})$, it is yet unknown
which point set is optimal in the sense that it is a global minimizer of the
worst case integration error. We will present a computer-assisted proof by
exhaustion that the Fibonacci lattice is the unique minimizer of the QMC worst
case error in periodic $H^1_\text{mix}$ for small $N$. Moreover, we investigate
the situation for pointsets whose cardinality $N$ is not a Fibonacci number. It
turns out that for $N=1,2,3,5,7,8,12,13$ the optimal point sets are integration
lattices.
