MacWilliams-type identities for fragment and sphere enumerators

Let P = n 1 1 ⊕ ⋯ ⊕ n t 1 be the poset given by the ordinal sum of the antichains n i 1 with n i elements. We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C ⊥ of the linear code C on P and those for C on the dual poset P ̌ . The linear changes of variables appearing in the identities are explicit. So we obtain, for example, the P -weight distribution of C ⊥ as the P ̌ -weight distribution times an invertible matrix which is a generalization of the Krawtchouk matrix.