Defining the $q$-Analogue of a Matroid

Jurrius, Relinde; Pellikaan, Ruud

doi:10.37236/6569

Cited by 30 publications

(127 citation statements)

References 14 publications

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“…Proof. The statement that P (Γ(C), c) = P (Γ ′ (C), c) follows from [9,Corollary 4.7]. The statement that P (Γ(C), r) ∼ P (Γ ′ (C), r) follows by Propositions 1.13 and 5.7.…”

Section: Structural Properties Of Codes Via Q-polymatroidsmentioning

confidence: 92%

“…It is known from [9] that a vector rank-metric code C ⊆ F n q m gives rise to a q-matroid M (C) on F n q . In our notation, M (C) = P (Γ(C), c), where Γ is any F q -basis of F q m .…”

Section: Structural Properties Of Codes Via Q-polymatroidsmentioning

confidence: 99%

“…Crapo [3] already studied this combinatorial object from the point of view of geometric lattices. Recently, Jurrius and Pellikaan [9] rediscovered q-matroids and associated a q-matroid to every vector rank-metric code. One goal of the current paper is extending this association to rank-metric codes.…”

Section: Introductionmentioning

confidence: 99%

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Rank-metric codes and q-polymatroids

et al. 2019

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This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.We start by establishing the notation and the definitions used throughout the paper.Notation 1.1. In the sequel, we fix integers n, m ≥ 2 and a prime power q. For an integer t, we let [t] := {1, ..., t}. We denote by F q the finite field with q elements. The space of n × m matrices with entries in F q is denoted by Mat. Up to transposition, we assume without loss of generality that n ≤ m. We let Mat(J, c) = {M ∈ Mat | colsp(M ) ⊆ J} and Mat(J, r) = {M ∈ Mat | rowsp(M ) ⊆ J}.Throughout the paper, we only consider linear codes. All dimensions are computed over F q , unless otherwise stated.

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“…Proof. The statement that P (Γ(C), c) = P (Γ ′ (C), c) follows from [9,Corollary 4.7]. The statement that P (Γ(C), r) ∼ P (Γ ′ (C), r) follows by Propositions 1.13 and 5.7.…”

Section: Structural Properties Of Codes Via Q-polymatroidsmentioning

confidence: 92%

“…It is known from [9] that a vector rank-metric code C ⊆ F n q m gives rise to a q-matroid M (C) on F n q . In our notation, M (C) = P (Γ(C), c), where Γ is any F q -basis of F q m .…”

Section: Structural Properties Of Codes Via Q-polymatroidsmentioning

confidence: 99%

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Rank-metric codes and q-polymatroids

et al. 2019

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“…2]), or in terms of anticodes [53], among others. The q-analog of a matroid has been recently introduced in [25], where its connection with linear codes in the case ℓ = 1 was given. Anticodes when ℓ = 1 were used in [48,53].…”

Section: Generalized Sum-rank Weightsmentioning

confidence: 99%

“…This is due to the following identities, which follow directly from Proposition 2. Observe that restricted and shortened codes are the key description in the matroidal approach to generalized weights (see [1,25]).…”

Section: Generalized Sum-rank Weightsmentioning

confidence: 99%

Theory of supports for linear codes endowed with the sum-rank metric

Martínez-Peñas

2019

Des. Codes Cryptogr.

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The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: 1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; 2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; 3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

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A polymatroid approach to generalized weights of rank metric codes

Ghorpade

Johnsen

2020

Des. Codes Cryptogr.

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We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.

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Defining the $q$-Analogue of a Matroid

Cited by 30 publications

References 14 publications

Rank-metric codes and q-polymatroids

Rank-metric codes and q-polymatroids

Theory of supports for linear codes endowed with the sum-rank metric

A polymatroid approach to generalized weights of rank metric codes

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