  Motivated by the discovery of new family 122 iron-based superconductors, we
present the theoretical results on the ground state phase diagram, spin wave
and dynamic structure factor of the extended $J_{1}-J_{2}$ Heisenberg model. In
the reasonable physical parameter region of $K_{2}Fe_{4}Se_{5}$, we fi{}nd the
block checkerboard antiferromagnetic order phase is stable. There are two
acoustic branches and six optical branches spin wave in the block checkerboard
antiferromagnetic phase, which has analytic expression in the high symmetry
points. To compare the further neutron scattering experiments, we discuss the
saddlepoint structure in the magnetic excitation spectrum and calculate the
predicted inelastic neutron scattering pattern based on linear spin wave
theory.
