Generalized Hamming Weights for Almost Affine Codes

Johnsen, Trygve; Verdure, Hugues

doi:10.1109/tit.2017.2654456

Cited by 24 publications

(27 citation statements)

References 18 publications

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“…We will use Proposition 3.5 as a point of departure for studying relative length/dimension proles of ags of almost ane codes in Subsection 3.2. The following result was also proved in [10], linking the denition of generalized Hamming weights for almost ane codes to an analogue of the "classical" denition of generalized Hamming weights for linear codes. Theorem 3.6.…”

Section: Denition 210 a Agmentioning

confidence: 88%

“…In [10,Proposition 4] one stated and proved the following result: Proposition 3.5. Let C be an almost ane code.…”

Section: Denition 210 a Agmentioning

confidence: 99%

“…for the matroid associated with (the rank function of) the dual code C * . In [10], we show that we can dene generalized Hamming weights for almost ane codes via its associated matroid, namely Denition 3.4. Let C be an almost ane code.…”

Section: Denition 210 a Agmentioning

confidence: 99%

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Relative profiles and extended weight polynomials of almost affine codes

Johnsen

Verdure

2018

Cryptogr. Commun.

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In this paper we study various aspects concerning almost ane codes, a class including, and strictly larger than, that of linear codes. We use the combinatorial tool demi-matroids to show how one can dene relative length/dimension and dimension/length proles of ags(chains) of almost ane codes. In addition we show two specic results. One such result is how one can express the relative length/dimension proles (also called relative generalized Hamming weights) of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can nd the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an innite hierarchy of extensions of a code over a given alphabet.

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Section: Denition 210 a Agmentioning

confidence: 88%

“…In [10,Proposition 4] one stated and proved the following result: Proposition 3.5. Let C be an almost ane code.…”

Section: Denition 210 a Agmentioning

confidence: 99%

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Relative profiles and extended weight polynomials of almost affine codes

Johnsen

Verdure

2018

Cryptogr. Commun.

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“…These numbers are a natural generalization of the notion of minimum distance and they have several applications from cryptography (codes for wire-tap channels of type II), t-resilient functions, trellis or branch complexity of linear codes, and shortening or puncturing structure of codes; see [1,4,6,10,11,12,17,20,25,28,29,31,32,33] and the references therein. If r = 1, we obtain the minimum distance δ(C) of C which is the most important parameter of a linear code.…”

Section: Introductionmentioning

confidence: 99%

Generalized Hamming weights of projective Reed–Muller-type codes over graphs

Martinez-Bernal

Valencia-Bucio

Villarreal

2020

Discrete Mathematics

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Let G be a connected graph and let X be the set of projective points defined by the column vectors of the incidence matrix of G over a field K of any characteristic. We determine the generalized Hamming weights of the Reed-Muller-type code over the set X in terms of graph theoretic invariants. As an application to coding theory we show that if G is non-bipartite and K is a finite field of char(K) = 2, then the r-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of G is the r-th weak edge biparticity of G. If char(K) = 2 or G is bipartite, we prove that the r-th generalized Hamming weight of that code is the r-th edge connectivity of G.2010 Mathematics Subject Classification. Primary 13P25; Secondary 94B05, 94C15, 11T71, 05C40.

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“…The generalized Hamming weights (GHWs) of a linear code are parameters of interest in many applications [12,16,20,27,31,37,42,43,45] and they have been nicely related to the graded Betti numbers of the ideal of cocircuits of the matroid of a linear code [19,20], to the nullity function of the dual matroid of a linear code [42], and to the enumerative combinatorics of linear codes [3,18,22,23]. Because of this, their study has attracted considerable attention, but determining them is in general a difficult problem.…”

Section: Introductionmentioning

confidence: 99%

Linear codes over signed graphs

Martinez-Bernal

Valencia-Bucio

Villarreal

2019

Des. Codes Cryptogr.

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We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual code. Then we determine the regularity of the ideals of circuits and cocircuits of a signed graph, and prove an algebraic formula in terms of the multiplicity for the frustration index of an unbalanced signed graph. IntroductionThe generalized Hamming weights (GHWs) of a linear code are parameters of interest in many applications [12,16,20,27,31,37,42,43,45] and they have been nicely related to the graded Betti numbers of the ideal of cocircuits of the matroid of a linear code [19,20], to the nullity function of the dual matroid of a linear code [42], and to the enumerative combinatorics of linear codes [3,18,22,23]. Because of this, their study has attracted considerable attention, but determining them is in general a difficult problem. The notion of generalized Hamming weight was introduced by Helleseth, Kløve and Mykkeltveit in [17] and was first used systematically by Wei in [42]. For convenience we recall this notion. Let K = F q be a finite field and let C be an [m, k]-linear code of length m and dimension k, that is, C is a linear subspace of K m with k = dim K (C). Let 1 ≤ r ≤ k be an integer. Given a linear subspace D of C, the support of D, denoted χ(D), is the set of nonzero positions of D, that is, χ(D) := {i : ∃ (a 1 , . . . , a m ) ∈ D, a i = 0}.The r-th generalized Hamming weight of C, denoted δ r (C), is given by δ r (C) := min{|χ(D)| : D is a subspace of C, dim K (D) = r}.As usual we call the set {δ 1 (C), . . . , δ k (C)} the weight hierarchy of the linear code C. The 1st Hamming weight of C is the minimum distance δ(C) of C, that is, one haswhere ω(x) is the Hamming weight of the vector x, i.e., the number of non-zero entries of x. To determine the minimum distance is essential to find good error-correcting codes [23].The notion of generalized Hamming weights for linear codes was extended to matroids by Britz, Johnsen, Mayhew and Shiromoto [6, p. 332] as we now explain.Let M be a matroid with ground set E, rank function ρ, nullity function η, and let M * be its dual matroid. The r-th generalized Hamming weight of M , denoted d r (M ), is given by d r (M ) := min{|X| : X ⊆ E and η(X) = r} for 1 ≤ r ≤ η(E).2010 Mathematics Subject Classification. Primary 94B05; Secondary 94C15, 05C40, 05C22; 13P25.

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Generalized Hamming Weights for Almost Affine Codes

Cited by 24 publications

References 18 publications

Relative profiles and extended weight polynomials of almost affine codes

Relative profiles and extended weight polynomials of almost affine codes

Generalized Hamming weights of projective Reed–Muller-type codes over graphs

Linear codes over signed graphs

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