Parameterized affine codes

López, Hiram H.; Sarmiento, Eliseo; Pinto, Maria Vaz; Villarreal, Rafael H.

doi:10.1556/sscmath.49.2012.3.1216

Cited by 9 publications

(16 citation statements)

References 16 publications

(15 reference statements)

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“…, A s of a field K, we denote the image of [21]. The basic parameters of the projective Reed-Muller-type code C X (d) are equal to those of C X * (d) [22]. A formula for the minimum distance of an affine cartesian code is given in [21, Theorem 3.8] and in [14,Proposition 5].…”

Section: Applications and Examplesmentioning

confidence: 99%

Minimum distance functions of complete intersections

2018

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Section: Applications and Examplesmentioning

confidence: 99%

Minimum distance functions of complete intersections

2018

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“…, y vs arising from the edges of a graph G or a clutter C (a clutter is a sort of hypergraph, see Definition 2.1). This paper is motivated by the study of parameterized linear codes [25], and specifically by the fact that the degree and the Hilbert function of S[u]/I(Y ) are related to the basic parameters of parameterized affine linear codes [22] (see Theorem 3.4).…”

Section: Introductionmentioning

confidence: 99%

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Pinto

Villarreal

2013

Communications in Algebra

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Let X * be a subset of an affine space A s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X * under the maps x → [x] and x → [(x, 1)] respectively, where [x] and [(x, 1)] are points in the projective spaces P s−1 and P s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y ). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X * is an affine torus if and only if I(Y ) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.2010 Mathematics Subject Classification. Primary 13F20; Secondary 13P25, 11T71, 94B25.

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“…where K * = K \ {0}. Following [15] we call X * an affine algebraic toric set parameterized by y v1 , . .…”

Section: Introductionmentioning

confidence: 99%

Vanishing Ideals Over Odd Cycles

Uribe-Paczka

Sarmiento

Márquez

2019

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let K be a finite field. Let X* be a subset of the a ne space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.

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Parameterized affine codes

Cited by 9 publications

References 16 publications

Minimum distance functions of complete intersections

Minimum distance functions of complete intersections

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Vanishing Ideals Over Odd Cycles

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