Generalized minimum distance functions

Sarabia, Manuel González; Martinez-Bernal, Jose; Villarreal, Rafael H.; Vivares, Carlos E.

doi:10.1007/s10801-018-0855-x

Cited by 20 publications

(26 citation statements)

References 46 publications

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“…By [14,Theorem 4.5] the r-th generalized Hamming weight of a projective Reed-Muller code is given by where F d,r the set of r-tuples of forms of degree d in S which are linearly independent over K modulo the ideal I and the maximum is taken to be 0 if F d,r = ∅.…”

Section: Introductionmentioning

confidence: 99%

“…As we can see above, the generalized Hamming weights for any linear code can be interpreted using the language of commutative algebra. Motivated by the notion of generalized Hamming weight described above and following [14] we define generalized minimum distance (GMD) functions for any homogeneous ideal in a polynomial ring. This allows us to extend the notion of generalized Hamming weights to codes arising from algebraic schemes, rather than just from reduced sets of points.…”

Section: Introductionmentioning

confidence: 99%

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Generalized minimum distance functions and algebraic invariants of Geramita ideals

Cooper

Seceleanu

Tohǎneanu

et al. 2020

Advances in Applied Mathematics

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Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function δI (d, r) of a graded ideal I in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that δI is non-decreasing as a function of r and non-increasing as a function of d. For vanishing ideals over finite fields, we show that δI is strictly decreasing as a function of d until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, 1-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Tohȃneanu-Van Tuyl and Eisenbud-Green-Harris.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Cooper

Seceleanu

Tohǎneanu

et al. 2020

Advances in Applied Mathematics

Self Cite

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“…In proving our result in this paper, we follow the footsteps of [11] and [1], where the results were derived using the notion of so-called footprint bound. Some early articles on footprint bounds include [8,14,10] and some recent articles include [19,13,2] among others. A somewhat brief discussion of the notion of the footprint bounds is given in Subsection 2.3.…”

Section: Introductionmentioning

confidence: 99%

Relative generalized Hamming weights of affine Cartesian codes

Datta

2020

Des. Codes Cryptogr.

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We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative generalized Hamming weights of Reed-Muller codes by Geil and Martin, as well as the determination of all the generalized Hamming weights of the affine Cartesian codes by Beelen and Datta. The author is supported by a postdoctoral fellowship from DST-RCN grant INT/NOR/RCN/ICT/P-03/2018.

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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%

“…The minimum distance of any linear code can be computed using SageMath [30]. For signed simple graphs one can also compute the minimum distance using Proposition 5.6 and the algorithms of [12,24]. For methods to calculate higher weight enumerators of linear codes see [5] and the references therein.…”

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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 minimum distance functions

Cited by 20 publications

References 46 publications

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Relative generalized Hamming weights of affine Cartesian codes

Linear codes over signed graphs

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