We improve Levenshtein's upper bound for the cardinality of a code of length four that is capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we construct an optimal perfect code that is capable of correcting single deletions for the same parameters.

In this paper, all poset structures that admit the MacWilliams identity are classified, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. It is proved that being a hierarchical poset is a necessary and sufficient condition for a poset to admit the MacWilliams identity. An explicit relation is also derived between the P-weight distribution of a hierarchical poset code and the P~-weight distribution of the dual code.