Recently, Etzion et al. introduced metrics on F2nbased on directed graphs on n vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code H̃3to be a two-perfect code. We also construct some families of such structures for any k ≥ 3, which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps.

Let A(n,d) (respectively A(n,d,w)) be the maximum possible number of codewords in a binary code (respectively, binary constant-weight w code) of length n and minimum Hamming distance at least d. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalizing the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on A(n,d), namely A(18,8) ≤ 71 and A(19,8) ≤ 131. Twenty three new upper bounds on A(n,d,w) for n ≤ 28 are also obtained by a similar way.

We give an alternative proof of Delsarte's linear programming bound for binary codes and its improvements. Applying the technique which is used in the proof to binary constant-weight codes, we obtain new upper bounds on sizes of binary constant-weight codes.

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.

A binary poset code of codimensionm(of cardinality 2n-m, wherenis the code length) can correct maximummerrors. All possible poset metrics that allow codes of codimensionmto bem-, (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 ofr-perfect poset codes for somerin the case of the crown poset and in the case of the union of disjoint chains.

A binary poset code of codimension m (of cardinality 2n-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.

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.