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Designs, Codes and Cryptography

Erschienen in:

30.11.2018

Cubic surfaces over small finite fields

verfasst von: Anton Betten, Fatma Karaoglu

Erschienen in: Designs, Codes and Cryptography | Ausgabe 4/2019

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Abstract

In the 1960s, Hirschfeld embarked on a program to classify cubic surfaces with 27 lines over finite fields. This work is a contribution to this problem. We develop an algorithm to classify surfaces with 27 lines over a finite field using the classical theory of double-sixes. This algorithm is used to classify these surfaces over all fields of order q at most 97. We then construct a family of cubic surfaces over finite fields of odd order. The generic surfaces in this family have six Eckardt points and they are invariant under a symmetric group of degree four. The family turns out to be isomorphic to the example of a family of cubic surface given over the real numbers by Hilbert and Cohn-Vossen.

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Titel: Cubic surfaces over small finite fields
verfasst von: Anton Betten
Fatma Karaoglu
Publikationsdatum: 30.11.2018
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 4/2019
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-018-0590-2

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