  The information reconstruction problem on an infinite tree, is to collect and
analyze massive data samples at the $n$th level of the tree to identify whether
there is non-vanishing information of the root, as $n$ goes to infinity. This
problem has wide applications in various fields such as biology, information
theory and statistical physics, and its close connections to cluster learning,
data mining and deep learning have been well established in recent years.
Although it has been studied in numerous contexts, the existing literatures
with rigorous reconstruction thresholds established are very limited. In this
paper, motivated by a classical deoxyribonucleic acid (DNA) evolution model,
the F$81$ model, and taking into consideration of the Chargaff's parity rule by
allowing the existence of a guanine-cytosine content bias, we study the noise
channel in terms of a $4\times 4$-state asymmetric probability transition
matrix with community effects, for four nucleobases of DNA. The corresponding
information reconstruction problem in molecular phylogenetics is explored, by
means of refined analyses of moment recursion, in-depth concentration
estimates, and thorough investigations on an asymptotic $4$-dimensional
nonlinear second order dynamical system. We rigorously show that the
reconstruction bound is not tight when the sum of the base frequencies of
adenine and thymine falls in the interval $\left(0,1/2-\sqrt{3}/6\right)\bigcup
\left(1/2+\sqrt{3}/6,1\right)$, which is the first rigorous result on
asymmetric noisy channels with community effects.
