MacWilliams-type identities for fragment and sphere enumerators

Let P=n11⊕⋯⊕nt1 be the poset given by the ordinal sum of the antichains ni1 with nini elements. We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C⊥C⊥ of the linear code CC on P and those for CC on the dual poset P̌. The linear changes of variables appearing in the identities are explicit. So we obtain, for example, the P-weight distribution of C⊥C⊥ as the P̌-weight distribution times an invertible matrix which is a generalization of the Krawtchouk matrix.