  A deep understanding of quantum entanglement is vital for advancing quantum
technologies. The strength of entanglement can be quantified by counting the
degrees of freedom that are entangled, which results in a quantity called
Schmidt number. A particular challenge is to identify the strength of
entanglement in quantum states which remain positive under partial transpose
(PPT), otherwise recognized as undistillable states. Finding PPT states with
high Schmidt number has become a mathematical and computational challenge. In
this work, we introduce efficient analytical tools for calculating the Schmidt
number for a class of bipartite states, called generalized grid states. Our
methods improve the best known bounds for PPT states with high Schmidt number.
Most notably, we construct a Schmidt number three PPT state in five dimensional
systems and a family of states with a Schmidt number of $(d+1)/2$ for odd
$d$-dimensional systems, representing the best-known scaling of the Schmidt
number in a local dimension. Additionally, these states possess intriguing
geometrical properties, which we utilize to construct indecomposable
entanglement witnesses.
