  We construct a large family of Planar Maximally Entangled (PME) states which
are a wider class of multi-partite entangled states than Absolutely Maximally
Entangled (AME) states. These are states in which any half of the qudits are in
a maximally mixed state, provided that they form a connected subset. We show
that in contrast to AMEs, PMEs are easier to find and there are various PMEs
for any even number of qudits. In particular, while it is known that no AME
state of four qubits exists, we show that there are two distinct
multi-parameter classes of four qubit PMEs. We also give explicit families of
PMEs for any even number of particles and for any dimension.
