Characterization of Metrics Induced by Hierarchical Posets

Machado, Roberto Assis; Pinheiro, Jerry Anderson; Firer, Marcelo

doi:10.1109/tit.2017.2691763

Cited by 24 publications

(18 citation statements)

References 25 publications

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“…(3) of Theorem 3.2 may be viewed as the canonical decomposition of the semi-simple code C. Assume that S = F is a finite field and H is a finite dimensional vector space over F. Then, with P set to be the canonical reduced form of a directed graph G, (3) of Theorem 3.2 recovers the "only if " parts of [10,Theorem 9] and [20,Theorem 1]. If we set H i = F for all i ∈ Ω, then (3) of Theorem 3.2 recovers [11, Corollary 1] and the "only if " part of [21,Theorem 2]. Now we consider some other sufficient conditions for H to satisfy the MacWilliams extension property with respect to P-support.…”

Section: Macwilliams Extension Property With Respect To P-support And...mentioning

confidence: 96%

“…Definition 1.1 is a variant of [20,Definition 7] and [10,Definition 2], and is crucial for studying the Fourier-reflexivity of the partition Q(H, P, ω) when H is finite, and furthermore, whether (Q(H, P, ω), Q(H, P, ω)) admits MacWilliams identity when H is a finite vector space. It might be worth noting that Definition 1.1 generalizes the well-known notion of a hierarchical poset (see [21]). Indeed, as a consequence of [21,Theorem 3] and the fact that I(P) = {Ω − I | I ∈ I(P)} (see [16,Lemma 1.2]), we have the following Lemma.…”

Section: Introductionmentioning

confidence: 99%

“…It might be worth noting that Definition 1.1 generalizes the well-known notion of a hierarchical poset (see [21]). Indeed, as a consequence of [21,Theorem 3] and the fact that I(P) = {Ω − I | I ∈ I(P)} (see [16,Lemma 1.2]), we have the following Lemma.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric

Xu¹,

Kan²,

Han³

2022

Preprint

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Let H be the cartesian product of a family of abelian groups indexed by a finite set Ω. A given poset P = (Ω, P ) and a map ω : Ω −→ R + give rise to the (P, ω)-weight on H, which further leads to a partition Q(H, P, ω) of H. For the case that H is finite, we give sufficient conditions for two codewords to belong to the same block of Λ, the dual partition of H, and sufficient conditions for H to be Fourier-reflexive. By relating the involved partitions with certain polynomials, we show that such sufficient conditions are also necessary if P is hierarchical and ω is integer valued. With H further set to be a finite vector space over a finite field F, from a partition perspective, we extend the property of "admitting MacWilliams identity" to arbitrary pairs of partitions of H, and prove that a pair of Finvariant partitions (Λ, Γ) with |Λ| = |Γ| admits MacWilliams identity if and only if (Λ, Γ) is a pair of mutually dual Fourier-reflexive partitions. Such a result is then applied to the partitions induced by P-weight and (P, ω)weight. With H set to be a (possibly infinite) left module over a ring S, we show that each (P, ω)-weight isometry of H uniquely induces an order automorphism of P, which further leads to a group homomorphism from the

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Section: Macwilliams Extension Property With Respect To P-support And...mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric

Xu¹,

Kan²,

Han³

2022

Preprint

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“…We will show that this is not the case for poset codes, that is, we may construct two codes, one perfect and the other not, both having the same parameters. This situation may occur for metrics determined by non-hierarchical posets, since for these metrics the minimum distance of a code does not determine its packing radius (see [18]). We start by characterizing the ones where the reciprocal is true.…”

Section: R-perfect Poset Codesmentioning

confidence: 99%

On perfect poset codes

Panek¹,

Pinheiro²,

Alves³

et al. 2020

AMC

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We consider on F n q metrics determined by posets and classify the parameters of 1-perfect poset codes in such metrics. We show that a code with same parameters of a 1-perfect poset code is not necessarily perfect, however, we give necessary and sufficient conditions for this to be true. Furthermore, we characterize the unique way up to a labeling on the poset, considering some conditions, to extend an r-perfect poset code over F n q to an r-perfect poset code over F n+m q .

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“…In the case the graph has no circuit, that is, in case G defines a poset, the so-called hierarchical posets play an exceptional role, since many of the known properties of codes, including MacWilliams Identity and Extension properties, hold for a poset metric if and only if the poset is hierarchical (see [12], [13], [6]). Many of those results, originally proved for the usual Hamming metric, depends essentially on the action of the group of linear isometries being transitive on spheres centered at 0.…”

Section: G-canonical Decomposition Of Linear Codes For Hierarchicmentioning

confidence: 99%

Metrics Based on Finite Directed Graphs and Coding Invariants

Etzion

Firer

Machado

2018

IEEE Trans. Inform. Theory

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Given a finite directed graph with n vertices, we define a metric d G on F n q , where F q is the finite field with q elements. The weight of a word is defined as the number of vertices that can be reached by a directed path starting at the support of the vector. Two canonical forms, which do not affect the metric, are given to each graph. Based on these forms we characterize each such metric. We further use these forms to prove that two graphs with different canonical forms yield different metrics. Efficient algorithms to check if a set of metric weights define a metric based on a graph are given. We provide tight bounds on the number of metric weights required to reconstruct the metric. Furthermore, we give a complete description of the group of linear isometries of the graph metrics and a characterization of the graphs for which every linear code admits a G-canonical decomposition. Considering those graphs, we are able to derive an expression of the packing radius of linear codes in such metric spaces. Finally, given a directed graph which determines a hierarchical poset, we present sufficient and necessary conditions to ensure the validity of the MacWilliams Identity and the MacWilliams Extension Property. i 0, then w(x) ≤ w(x ′ ).

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Characterization of Metrics Induced by Hierarchical Posets

Cited by 24 publications

References 25 publications

Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric

Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric

On perfect poset codes

Metrics Based on Finite Directed Graphs and Coding Invariants

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