Classification of poset-block spaces admitting MacWilliams-type identity

TL;DR

This paper characterizes poset-block spaces that admit MacWilliams-type identities, showing they must be hierarchical with uniform block dimensions at each level, and provides explicit relations between code and dual weight enumerators.

Contribution

It establishes necessary and sufficient conditions for poset-block spaces to admit MacWilliams-type identities and derives explicit formulas for weight enumerator relations.

Findings

Poset-block spaces admit MacWilliams-type identities iff the poset is hierarchical with uniform block dimensions.

Explicit formulas relate weight enumerators of codes and their duals in these spaces.

Provides a complete characterization of such spaces based on poset structure.

Abstract

In this work we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity we explicit the relation between the weight enumerators of a code and its dual.