  Permutation codes under different metrics have been extensively studied due
to their potentials in various applications. Generalized Cayley metric is
introduced to correct generalized transposition errors, including previously
studied metrics such as Kendall's $\tau$-metric, Ulam metric and Cayley metric
as special cases. Since the generalized Cayley distance between two
permutations is not easily computable, Yang et al. introduced a related metric
of the same order, named the block permutation metric. Given positive integers
$n$ and $d$, let $\mathcal{C}_{B}(n,d)$ denote the maximum size of a
permutation code in $S_n$ with minimum block permutation distance $d$. In this
paper, we focus on the theoretical bounds of $\mathcal{C}_{B}(n,d)$ and the
constructions of permutation codes under block permutation metric. Using a
graph theoretic approach, we improve the Gilbert-Varshamov type bound by a
factor of $\Omega(\log{n})$, when $d$ is fixed and $n$ goes into infinity. We
also propose a new encoding scheme based on binary constant weight codes.
Moreover, an upper bound beating the sphere-packing type bound is given when
$d$ is relatively close to $n$.
