Combinatorial metrics: MacWilliams-type identities, isometries and extension property

A tiling of the n-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the n-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the concatenation of two smaller ones. I. INTRODUCTIONPerfect codes have been extensively studied in the literature, due to the optimality condition imbued in its definition and to the interesting challenges they pose. Nevertheless, considering the Hamming metric, perfect codes are rare and they are classified in the case of linear binary codes: It has the same parameters of binary repetition codes with odd length, q-ary Hamming codes, binary or ternary Golay codes, [16], [21]. The situation is not the same for the Lee metric. There are few results (see [17], [18], [33]), but there is not a complete characterization. Two good surveys about perfect codes in Hamming metric are [22], [28].The most primary definition of a perfect code is the geometrical one: a code is perfect if its packing radius equals its covering radius. This definition is interesting in our setting since it depends directly on the metric invariants, which are naturally defined in general settings. To be more explicit, given a metric space (X, d) and a subset C ⊂ X, we define its packing radius R d,pack (C) as the maximal r such the balls of radius r centered at elements of C are disjoint and its covering radius R d,cov (C) as the minimum r such that the balls of radius r centered at elements of C covers the space X. The set C is called a d-perfect code if R d,pack (C) = R d,cov (C). Set this, it is understandable that the study of perfect codes can be undertaken considering a more vast family of metrics and it can be valuable to do so for any metric that has some relevance in the context of coding theory.Considering this scarcity of perfect codes under the Hamming metric, the introduction of the poset metrics by Brualdi et al. in 1995 [6] drawn the attention since, in general, there is a relative abundance of perfect codes (depending on the poset metric). The study of perfect codes in this context is done in different approaches. One of them is to fix a particular family of posets (chain, crown or hierarchical) and to classify all the perfect codes for the particular family, as done in [25], [26], [31].Another approach is to consider a family of well known codes and asks to classify all the poset metrics which turn the code to be perfect. This is what is done, for example, with the extended Hamming and Golay codes for poset metrics (in [23], [24]) and for poset-block metrics ( [2], [8]). Our approach resembles more the second one, but instead of looking at the ...

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“…2) Combinatorial metric: The combinatorial metrics were introduced by Gabidulin in [15]. For more details see [4], [12], [32].…”

Section: Definitionmentioning

confidence: 99%

Obtaining Binary Perfect Codes Out of Tilings

Miyamoto

Firer

2020

IEEE Trans. Inform. Theory

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“…Recently, the combinatorial metrics started to be explored in a more systematic way. In [71] the authors give a necessary and sufficient condition for the existence of a MacWilliams Identity, that is, conditions of F to ensure that the…”

Section: Combinatorial Metricsmentioning

confidence: 99%

Alternative Metrics

Firer

2019

Preprint

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“…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. This property has been first introduced by Kim and Oh in [18], where the authors have proven that being hierarchical is a necessary and sufficient condition for a poset to admit MacWilliams identity. The original property has since been extended and generalized to poset block metric by Pinheiro and Firer in [29], to combinatorial metric by Pinheiro, Machado and Firer in [30], and to directed graph metric by Etzion, Machado and Firer in [10]. In [8], Choi, Hyun, Kim and Oh have proposed and studied MacWilliams-type relations, which, roughly speaking, are defined as equivalent relations on I(P) which admit MacWilliams identities (see [8, Definitions 2.1 and 2.7] for more details).…”

Section: Introductionmentioning

confidence: 99%

“…With (H i | i ∈ Ω) set to be a family of (possibly infinite) left modules over a ring S, we study the group of (P, ω)-weight isometries, i.e., S-module automorphisms of H that preserves (P, ω)-weight. Groups of linear isometrics have been characterized for crown-weight by Cho and Kim in [7], for poset metric by Panek, Firer, Kim and Hyun in [27], for poset block metric by Alves, Panek and Firer in [1], for directed graph metric by Etzion, Firer and Machado in [10], for combinatorial metric by Pinheiro, Machado and Firer in [30], and for general additive metrics by Panek and Pinheiro in [28].…”

Section: Introductionmentioning

confidence: 99%

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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Combinatorial metrics: MacWilliams-type identities, isometries and extension property

Abstract: In this work we characterize the combinatorial metrics admitting a MacWilliams-type identity and describe the group of linear isometries of such metrics. Considering coverings that are not connected, we classify the metrics satisfying the MacWilliams extension property.

Cited by 10 publications

References 10 publications

Obtaining Binary Perfect Codes Out of Tilings

Obtaining Binary Perfect Codes Out of Tilings

Alternative Metrics

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

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