The Poset Metrics That Allow Binary Codes of Codimension $m$ to be $m$ -, $(m-1)$-, or $(m-2)$-Perfect

A binary poset code of codimension
m
(of cardinality 2
n
-
m
, where
n
is the code length) can correct maximum
m
errors. All possible poset metrics that allow codes of codimension
m
to be
m
-, (
m
-1)-, or (
m
-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of
r
-perfect poset codes for some
r
in the case of the crown poset and in the case of the union of disjoint chains.