  This paper focuses on $ K $-receiver discrete-time memoryless broadcast
channels (DM-BCs) with private messages, where the transmitter wishes to convey
$K$ private messages to $K$ receivers respectively. A general inner bound on
the capacity region is proposed based on an exhaustive message splitting and a
$K$-level modified Marton's coding. The key idea is to split every message into
$ \sum_{j=1}^K {K\choose j} $ submessages each corresponding to a set of users
who are assigned to recover them, and then send these submessages through
codewords that are jointly typical with each other. To guarantee the joint
typicality among all transmitted codewords, a sufficient condition on the
subcodebooks sizes is derived through a newly establishing hierarchical
covering lemma, which extends the 2-level multivariate covering lemma to the
$K$-level case including $(2^{K}-1)$ random variables with more intricate
dependence. As the number of auxiliary random variables and rate constraints
both increase linearly with $(2^{K}-1)$, the standard Fourier-Motzkin
elimination procedure becomes infeasible when $K$ is large. To tackle this
problem, we obtain the final form of achievable rate region with a special
observation of disjoint unions of sets that constitute the power set of $
\{1,\dots,K\}$. The proposed achievable rate region allows arbitrary input
probability mass functions (pmfs) and improves over all previously known ones
for $ K$-receiver ($K\geq 3$) BCs whose input pmfs should satisfy certain
Markov chain(s).
