Generalized weights: An anticode approach

Ravagnani, Alberto

doi:10.1016/j.jpaa.2015.10.009

Cited by 43 publications

(106 citation statements)

References 12 publications

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“…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]. Analogous reinterpretations of generalized sum-rank weights are left open.…”

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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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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“…Proof. By [17,Lemma 27.2], Γ(C) ∼ Γ ′ (C). Hence we may assume without loss of generality that Γ = Γ ′ = {γ 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…More definitions are due to Jurrius and Pellikaan [8] and Martínez-Peñas and Matsumoto [10], who also compared the various definitions. Using the theory of anticodes, Ravagnani [17] gave a definition of generalized weights for matrix rank-metric codes, which extends the one from [12].…”

Section: Introductionmentioning

confidence: 99%

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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“…extended in [21] and [17] to codes that are linear over the base field, where they are called Delsarte generalized weights and generalized matrix weights, respectively. We will use the term GRWs for the latter parameters, which were also found to measure the worst-case information leakage for codes that are linear over the base field [17,Th.…”

Section: Vii-b])mentioning

confidence: 99%

“…In this subsection, we will compare the codes C red ⊆ F We next argue the advantages of the codes C red over the codes C T Gab : 1) Generalized rank weights: Although GRWs have recently been extended to F q -linear codes [21], [17] and its connection to worst-case information leakage has been obtained [17,Th. 3], little is known about them for codes that are not linear over F q m .…”

Section: Comparison With Other Mrd Codesmentioning

confidence: 99%

Generalized rank weights of reducible codes, optimal cases and related properties

Martínez-Peñas

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error-correction in network coding. In this paper, we study their security behaviour against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst-case information leakage to the wire-tapper, 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal), and use the previous bounds to estimate their higher GRWs, 3) we show that all linear (over the extension field) codes whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence, and 4) we show that the information leaked to a wire-tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: Conditions to be rank equivalent to cartesian products of linear codes, conditions to be rank degenerate, duality properties and MRD ranks.

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Generalized weights: An anticode approach

Cited by 43 publications

References 12 publications

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

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

Rank-metric codes and q-polymatroids

Generalized rank weights of reducible codes, optimal cases and related properties

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