Minimum weight codewords in dual algebraic-geometric codes from the Giulietti-Korchmáros curve

Bartoli, Daniele; Bonini, Matteo

doi:10.1007/s10623-018-0541-y

Cited by 9 publications

(9 citation statements)

References 27 publications

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“…The last case we have to deal with is the one in which S 1 has exactly four singular points P 1 , P 2 , P 3 and P 4 . Note that if one among P 1 , P 2 , P 3 and P 4 is F q -rational, then we already have the desired bound given by (5). If instead the four points are two couples of conjugates F q 2 -rational points, then the bound is already established by (6).…”

Section: Four Singular Pointsmentioning

confidence: 91%

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Rational points on cubic surfaces and AG codes from the Norm-Trace curve

Bonini,

Sala,

Vicino

2021

Preprint

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In this paper we give a complete characterization of the intersections between the Norm-Trace curve over F q 3 and the curves of the form y = ax 3 + bx 2 + cx + d, generalizing a previous result by Bonini and Sala, providing more detailed information about the weight spectrum of one-point AG codes arising from such curve. We also derive, with explicit computations, some general bounds for the number of rational points on a cubic surface defined over Fq.

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Section: Four Singular Pointsmentioning

confidence: 91%

“…For AG codes, it is possible to derive information about their weight spectrum by the study of the intersection of the base curve X and low degree curves, as done in [3,5,12,28,29].…”

mentioning

confidence: 99%

Rational points on cubic surfaces and AG codes from the Norm-Trace curve

Bonini,

Sala,

Vicino

2021

Preprint

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“…Several papers have been dedicated to the construction of AG codes from the GK curves; see [1,2,4,7]. The GK-curves are q 6-maximal curves due to Giulietti and Korchmáros, which provided the first family of maximal curves that are not subcovers of the Hermitian curve [9].…”

Section: Introductionmentioning

confidence: 99%

AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

Lia

Timpanella

2021

AAECC

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In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve $${\mathcal {X}}$$ X were investigated and the sets of minimal generators were determined for all points in $${\mathcal {X}}(\mathbb {F}_{q^2})$$ X ( F q 2 ) and $${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$ X ( F q 6 ) \ X ( F q 2 ) . This paper completes their work by settling the remaining cases, that is, for points in $${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$ X ( F ¯ q ) \ X ( F q 6 ) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in $${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$ X ( F q 7 ) \ X ( F q ) and we give a bound on the Feng–Rao minimum distance $$d_{ORD}$$ d ORD . For $$q=3$$ q = 3 we provide a table that also reports the exact values of $$d_{ORD}$$ d ORD . As a further application we construct quantum codes from $$\mathbb {F}_{q^7}$$ F q 7 -rational points of the GK-curve.

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“…This led to the question whether every maximal curve is a subcover of the Hermitian curve or not. This question has a negative answer: in [13], Giulietti and Korchmáros introduced an infinity family of curves C ′ , the so called GK curve, which is maximal over F q 6 . Codes from the GK curve have been widely investigate, see for example [6,7,10,12] In most cases, the weight distribution of a given code is hard to be computed.…”

Section: Introductionmentioning

confidence: 99%

“…This question has a negative answer: in [13], Giulietti and Korchmáros introduced an infinity family of curves C ′ , the so called GK curve, which is maximal over F q 6 . Codes from the GK curve have been widely investigate, see for example [6,7,10,12] In most cases, the weight distribution of a given code is hard to be computed. Even the problem of computing codewords of minimum weight can be a difficult task apart from specific cases.…”

Section: Introductionmentioning

confidence: 99%