MacWilliams extension theorems and the local–global property for codes over Frobenius rings

Abstract

The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local-global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom-Tsfasman weight for vector codes satisfies the extension theorem, whereas the Niederreiter-Rosenbloom-Tsfasman weight for matrix codes does not. A short character-theoretic proof of the well-known MacWilliams extension theorem for the homogeneous weight is provided. Moreover it is shown that the extension theorem carries over to direct products of weights, but not to symmetrized products.

Characterization of Metrics Induced by Hierarchical Posets

“…The subject has been studied in many different contexts, and several papers were devoted to this subject, including [6], [20] and [21]. preserving the P -distance, there is a linear isometry T ∈ GL P (F q ) such that T | C = t.…”

“…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.…”

The extension problem for Lee and Euclidean weights

“…For this weight, we give a simple proof that works for all Z/N Z. This proof is similar to the proof in [2,Theorem 6.6] for the homogeneous weight over any finite Frobenius ring.…”

Metrics Based on Finite Directed Graphs and Coding Invariants

“…The condition established in Theorem 9 for a graph G(V, E) to admit a Gdecomposition is not sufficient to ensure the MacWilliams properties. Indeed, consider the metric induced by the graph G( [4], E = {(3, 4), (4, 3)}). Let C 1 = {0000, 1100} and C 2 = {0000, 0011} be two one-dimensional binary codes.…”