  Absence of magnetic ordering in Bi$_3$Mn$_4$O$_{12}$(NO$_3$), (BMNO) which
has a magnetic subsystem that consists of honeycomb bi-layers of Mn$^{4+}$ ions
with spin S=3/2, has raised the expectation that its ground state is strongly
frustrated due to longer-range antiferromagnetic interactions. We calculate the
magnetic exchange coupling constants of the BMNO complex within a density
functional approach and find that the dominating interactions between Mn spins
are the antiferromagnetic nearest-neighbor $J_1$ and interlayer interaction
$J_c$. The largest interaction is $J_c$, which substantially exceeds $J_1$.
Longer-range interactions are antiferromagnetic, but only weakly frustrating.
