  Absolutely maximally entangled (AME) states of $k$ qudits (also known as
perfect tensors) are quantum states that have maximal entanglement for all
possible bipartitions of the sites/parties. We consider the problem of whether
such states can be decomposed into a tensor network with a small number of
tensors, such that all physical and all auxiliary spaces have the same
dimension $D$. We find that certain AME states with $k=6$ can be decomposed
into a network with only three 4-leg tensors; we provide concrete solutions for
local dimension $D=5$ and higher. Our result implies that certain AME states
with six parties can be created with only three two-site unitaries from a
product state of three Bell pairs, or equivalently, with six two-site unitaries
acting on a product state on six qudits. We also consider the problem for
$k=8$, where we find similar tensor network decompositions with six 4-leg
tensors.
