Metrics Based on Finite Directed Graphs and Coding Invariants

Abstract

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 ′ ).

Weights which respect support and NN-decoding

“…In this section, we introduce a particular family which is a generalization for the digraph metrics which arises naturally from the reduced canonical form for directed graphs presented in [6]. This canonical form makes a contraction of each maximal cycle into a unique vertex and, then, such vertex is labeled by the number of vertices contained in the original cycle.…”

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

“…In the context of coding theory, the linear group of isometries has been characterized considering many different metrics (see for example [11,3]) and been used as a relevant tool to prove coding related results (see [4,7]). We aim to characterize the group of linear isometries of a space endowed with a combinatorial metric.…”

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

“…On the other hand, with P set to be an anti-chain, (1.2) recovers the notion of weight Hamming metric (see [5]). Furthermore, weighted poset metric can be viewed as an algebraic version of directed graph metric introduced by Etzion, Firer and Machado in [10] (see [17, Section I, Paragraphs 1-2]). It has been shown in [17, Section 2] (for binary field alphabet) that these two metrics are "almost the same".…”

“…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).…”

“…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].…”