A Classification of Posets Admitting the MacWilliams Identity

Abstract

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.

Characterization of Metrics Induced by Hierarchical Posets

“…In the case of poset metrics, to attain such a relation one needs to consider the dual poset: given a poset P = ([n], P ), the dual poset is the poset P = ([n], P ) defined by the opposite relations, i.e., given i, j ∈ [n], then For poset metrics, the MacWilliams Identity was first investigated in [19]. In [4] it was proved that the hierarchical posets are the only posets admitting a MacWilliams' identity, furthermore, explicit formulae for those identities was obtained.…”

“…Association schemes, which is a classical structure studied in algebraic combinatorics, came into the picture in [5] in order to provide a proof of the MacWilliams identity which is more direct than the one given in [4].…”

“…Before we proceed to the proof, we recall that some of those statements are known: items P 1 , P 2 , P 3 and P 4 are proved in [4], [6], [5] and [24]; item P 5 is partially proved in [7]. Nevertheless, we do present a complete proof for each of these statements, since the P -canonical decomposition allows us to do it in an elegant and short way.…”

“…Over the years, the following code-related properties were proven to hold when considering a metric determined by a hierarchical poset: (i) the weight enumerator of a code is completely determined by the weight enumerator of its dual code (MacWilliams-type Identity), [4]; (ii) a linear code determines an association scheme, [5]; (iii) isometric linear isomorphism between codes may be extended to the entire space (MacWilliams Extension Theorem), [6]; (iv) the packing radius of a code is a function of its minimum distance, [7]. These properties appear dispersed throughout the literature and were proved by using many different combinatorial and algebraic tools: characters, association schemes, matroids, etc.…”

Poset Metrics Admitting Association Schemes and a New Proof of Macwilliams Identity

“…The latter is equivalent to the poset metric over the disjoint union of chains of the same cardinality. Many subsequent attempts have been made to find the MacWilliams-type identities about linear codes for such metrics corresponding to the MacWilliams identity about linear codes for the Hamming distance (see [8,9,10,11,12,20]). …”