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

Matteo Bonini; Massimiliano Sala; Lara Vicino

doi:10.1007/s10231-022-01237-3

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

Matteo Bonini
, Massimiliano Sala
, Lara Vicino^*

^*Kontaktforfatter

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

1 Citationer (Scopus)

93 Downloads (Pure)

Abstract

In this paper, we derive general bounds for the number of rational points on a cubic surface defined over F_q, which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm–Trace curve over Fq3 and the curves of the form y= ax³+ bx²+ cx+ d, generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point AG codes arising from such curve.

Originalsprog	Engelsk
Tidsskrift	Annali di Matematica Pura ed Applicata
Vol/bind	202
Sider (fra-til)	185-208
Antal sider	24
ISSN	0373-3114
DOI	https://doi.org/10.1007/s10231-022-01237-3
Status	Udgivet - feb. 2023

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Publisher Copyright:
© 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

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10.1007/s10231-022-01237-3

Norm_Trace_CubicsAccepteret manuskript, 339 KB

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title = "Rational points on cubic surfaces and AG codes from the Norm{\textendash}Trace curve",

abstract = "In this paper, we derive general bounds for the number of rational points on a cubic surface defined over Fq, which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm{\textendash}Trace curve over Fq3 and the curves of the form y= ax3+ bx2+ cx+ d, generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point AG codes arising from such curve.",

keywords = "AG code, Cubic surfaces, Norm{\textendash}Trace curve, Weight spectrum",

author = "Matteo Bonini and Massimiliano Sala and Lara Vicino",

note = "Funding Information: The authors would like to thank the anonymous referee for their helpful and constructive comments that highly contributed to improve and generalize the results of the paper. The research of M. Bonini was supported by the Irish Research Council, Grant No. GOIPD/2020/597. The results showed in this paper are included in L. Vicino{\textquoteright}s MSc thesis (supervised by the first and the second author). Publisher Copyright: {\textcopyright} 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2023",

month = feb,

doi = "10.1007/s10231-022-01237-3",

language = "English",

volume = "202",

pages = "185--208",

journal = "Annali di Matematica Pura ed Applicata",

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TY - JOUR

T1 - Rational points on cubic surfaces and AG codes from the Norm–Trace curve

AU - Bonini, Matteo

AU - Sala, Massimiliano

AU - Vicino, Lara

N1 - Funding Information: The authors would like to thank the anonymous referee for their helpful and constructive comments that highly contributed to improve and generalize the results of the paper. The research of M. Bonini was supported by the Irish Research Council, Grant No. GOIPD/2020/597. The results showed in this paper are included in L. Vicino’s MSc thesis (supervised by the first and the second author). Publisher Copyright: © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/2

Y1 - 2023/2

N2 - In this paper, we derive general bounds for the number of rational points on a cubic surface defined over Fq, which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm–Trace curve over Fq3 and the curves of the form y= ax3+ bx2+ cx+ d, generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point AG codes arising from such curve.

AB - In this paper, we derive general bounds for the number of rational points on a cubic surface defined over Fq, which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm–Trace curve over Fq3 and the curves of the form y= ax3+ bx2+ cx+ d, generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point AG codes arising from such curve.

KW - AG code

KW - Cubic surfaces

KW - Norm–Trace curve

KW - Weight spectrum

UR - https://www.scopus.com/pages/publications/85133248339

U2 - 10.1007/s10231-022-01237-3

DO - 10.1007/s10231-022-01237-3

M3 - Journal article

AN - SCOPUS:85133248339

SN - 0373-3114

VL - 202

SP - 185

EP - 208

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

ER -

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

Abstract

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