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Applicable Algebra in Engineering, Communication and Computing
Published in: Applicable Algebra in Engineering, Communication and Computing 4/2021

02-01-2020 | Original Paper

Explicit maximal and minimal curves of Artin–Schreier type from quadratic forms

Authors: Daniele Bartoli, Luciane Quoos, Zülfükar Saygı, Emrah Sercan Yılmaz

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 4/2021

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Abstract

In this note we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin–Schreier type and the construction is closely related to quadratic forms from \({\mathbb {F}}_{q^n}\) to \({\mathbb {F}}_q\).
previous article On the algebraic structure of and applications to cryptography
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Literature
1.
Anbar, N., Meidl, W.: More on quadratic functions and maximal Artin–Schreier curves. Appl. Algebra Eng. Commun. Comput. 26(5), 409–426 (2015)MathSciNetCrossRef
2.
Anbar, N., Meidl, W.: Quadratic functions and maximal Artin–Schreier curves. Finite Fields Appl. 30, 49–71 (2014)MathSciNetCrossRef
3.
Çakçak, E., Özbudak, F.: Some Artin–Schreier type function fields over finite fields with prescribed genus and number of rational places. J. Pure Appl. Algebra 210(1), 113–135 (2007)MathSciNetCrossRef
4.
Fitzgerald, R.W.: Highly degenerate quadratic forms over finite fields of characteristic 2. Finite Fields Appl. 11, 165–181 (2005)MathSciNetCrossRef
5.
Fitzgerald, R.W.: Highly degenerate quadratic forms over over \({{\mathbb{F}}}_2\). Finite Fields Appl. 13, 778–792 (2007)MathSciNetCrossRef
6.
Garcia, A., Stichtenoth, H., Xing, C.: On subfields of the Hermitian function field. Compos. Math. 120, 137–170 (2000)MathSciNetCrossRef
7.
Güneri, C.: Artin-Schreier curves and weights of two-dimensional cyclic codes. Finite Fields Appl. 10(4), 481–505 (2004)MathSciNetCrossRef
8.
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)MATH
9.
McGuire, G., Yılmaz, E.S.: Divisibility of L-polynomials for a family of Artin–Schreier curves. J. Pure Appl. Algebra 223(8), 3341–3358 (2019)MathSciNetCrossRef
10.
Niederreiter, H., Xing, C.: Rational Points on Curves over Finite Fields: Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRef
11.
Niederreiter, H., Xing, C.: Algebraic Geometry in Coding Theory and Cryptography. Princeton University Press, Princeton (2009)CrossRef
12.
Özbudak, F., Saygı, E., Saygı, Z.: Quadratic forms of codimension 2 over certain finite fields of even characteristic. Cryptogr. Commun. 3, 241–257 (2011)MathSciNetCrossRef
13.
Özbudak, F., Saygı, E., Saygı, Z.: Quadratic forms of codimension 2 over finite fields containing \({{\mathbb{F}}}_4\) and Artin–Schreier type curves. Finite Fields Appl. 18, 396–433 (2012)MathSciNetCrossRef
14.
Özbudak, F., Saygı, Z.: On the number of quadratic forms having codimension 2 radicals in characteristic 2 giving maximal/minimal curves. Commun. Algebra 42(9), 3795–3810 (2014)MathSciNetCrossRef
15.
Özbudak, F., Saygı, Z.: Explicit maximal and minimal curves over finite fields of odd characteristics. Finite Fields Appl. 42, 81–92 (2016)MathSciNetCrossRef
16.
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Berlin (2009)MATH
17.
Tsfasman, M.A., Vladut, S.G., Nogin, D.: Algebraic Geometric Codes: Basic Notions. American Mathematical Society, Providence (2007)CrossRef
18.
Van Der Geer, G., Van Der Vlugt, M.: Reed–Muller codes and supersingular curves. I. Compos. Math. 84(3), 333–367 (1992)MathSciNetMATH
Metadata
Title
Explicit maximal and minimal curves of Artin–Schreier type from quadratic forms
Authors
Daniele Bartoli
Luciane Quoos
Zülfükar Saygı
Emrah Sercan Yılmaz
Publication date
02-01-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 4/2021
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-019-00411-0

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