In this paper all poset structures are classi£ed which admit the MacWilliams identity, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. We prove that being a hierarchical poset is a necessary and suf£cient condition for a poset to admit MacWilliams identity. An explicit relation is also derived between P-weight distribution of a hierarchical poset code and P-weight distribution of the dual code.

A (w, r ) cover-free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A binary (w, r ) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as a concept of key distribution patterns. In this paper, we develop a method of constructing superimposed codes and prove that some superimposed codes constructed in this way are optimal.

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