  Permutation codes have received a great attention due to various
applications. For different applications, one needs permutation codes under
different metrics. The generalized Cayley metric was introduced by Chee and Vu
[4] and this metric includes several other metrics as special cases. However,
the generalized Cayley metric is not easily computable in general. Therefore
the block permutation metric was introduced by Yang et al. [22] as the
generalized Cayley metric and the block permutation metric have the same
magnitude. However, the block permutation metric lacks the symmetry property
which restricts more advanced algebraic tools to be involved. In this paper, by
introducing a novel metric closely related to the block permutation metric, we
build a bridge between some advanced algebraic methods and codes in the block
permutation metric. More specifically, based on some techniques from algebraic
function fields originated in [19], we give an algebraic-geometric construction
of codes in the novel metric with reasonably good parameters. By observing a
trivial relation between the novel metric and block permutation metric, we then
produce non-systematic codes in block permutation metric that improve all known
results given in [21, 22]. More importantly, based on our non-systematic codes,
we provide an explicit and systematic construction of codes in block
permutation metric which improves the systematic result shown in [22]. In the
end, we demonstrate that our codes in the novel metric itself have reasonably
good parameters by showing that our construction beats the corresponding
Gilbert-Varshamov bound.
