Canonical- systematic form for codes in hierarchical poset metrics

Characterization of Metrics Induced by Hierarchical Posets

“…Proof: If P is hierarchical, then the existence of a P -canonical decomposition follows from the construction just described, for more details, see [7,Corollary 1]. Suppose P is not hierarchical and let i ∈ [l] be the lowest level of P for which there are a ∈ Γ i and b ∈ Γ i+1 such that a b, i.e., b is not greater than a according to P .…”

“…When considering metrics determined by hierarchical posets, it was shown in [7] that generator matrices can be characterized by a form similar to the standard one, known as canonicalsystematic form. The important part of this characterization is the canonical part, so we refer to it as the canonical form of a code.…”

“…By using the P -canonical decomposition, in [7], the authors gave explicit formulae for the minimum distance and the packing radius. We will describe those results and provide explicit formulae for the remaining invariants, the covering and Chebyshev radii.…”

“…For all those properties (including the known ones) we give simple and short proofs. The proofs are based on the existence of a canonical decomposition introduced by Felix and Firer [7] and on a simple counterexample.…”

“…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.…”