For some perturbed $Z_2-$(or $Z_4-)$equivariant planar Hamiltonian vector field sequnces of degree $n\ (n=2^k-1$ and $ n=3\times 2^{k-1}-1, k=2,3,\cdots),$ some new lower bounds for $H(n)$ in Hilbert's 16th problem and configurations of compound eyes of limit cycles are given, by using the bifurcation theory of planar dynamical systems and the quadruple transformation method given by Christopher and Lloyd. It gives rise to more exact results than Ref.[6].