AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

Montanucci, Maria; Timpanella, Marco; Zini, Giovanni

doi:10.1007/s00022-018-0428-0

Cited by 14 publications

(8 citation statements)

References 45 publications

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“…Among all the classical codes used to produce quantum stabilizer codes, Algebraic-Geometry (AG) codes [14] have received considerable attention [2,8,10,19,[23][24][25][26][27][28][29][31][32][33]41]. (One may also ask whether AG codes can be used also in relation to non-stabilizer quantum codes.…”

Section: Introductionmentioning

confidence: 99%

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

Bartoli

Montanucci

Zini

2021

Des. Codes Cryptogr.

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In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of wellknown algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón-Bezerra-Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández.

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Section: Introductionmentioning

confidence: 99%

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

Bartoli

Montanucci

Zini

2021

Des. Codes Cryptogr.

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“…AG codes are proven to have good performances provided that X , G and D are carefully chosen in an appropriate way. In particular, AG codes with better parameters can arise from curves which have many q -rational points, especially from maximal curves which are curves defined over q with q square whose number of q -rational points X( q ) attains the Hasse-Weil upper bound, namely �X( q )� = q + 1 + 2 √ q , where is the genus of X ; for AG codes from maximal curves see for instance [6,13,17,18]. Regarding the choice of the two divisors D and G, the latter is typically taken to be a multiple mP of a single point P of degree one.…”

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 method, known as CSS construction, has allowed to find many powerful quantum stabilizer codes. Among all the classical codes used to produce quantum stabilizer codes, Algebraic-Geometry (AG) codes [12] have received considerable attention [2,6,8,13,[17][18][19][20][21][22][23][24][25][26]29]. The interest towards AG codes is due to several reasons.…”

Section: Introductionmentioning

confidence: 99%

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

Bartoli¹,

Montanucci²,

Zini³

2019

Preprint

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In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón-Bezerra-Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández.

show abstract

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

Cited by 14 publications

References 45 publications

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

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

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

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