  Our study focuses on constructing optimal rank-1 lattices that enable exact
integration and reconstruction of functions in the Chebyshev space, based on
finite index sets. We introduce novel theoretical lower bounds on the minimum
number of integrands needed for reconstruction, show equivalence between
different plans for reconstruction under certain conditions and propose an
innovative algorithm for generating the optimal generator vector of rank-1
lattices. By leveraging the inherent structure of the set of interpolators, our
approach ensures admissibility conditions through exhaustive search and
verification, outperforming existing methods in terms of computation time and
memory usage. Numerical experiments validate the efficiency and practical
applicability of our algorithm.
