More on quadratic functions and maximal Artin-Schreier curves
type: Item
instance of: scholarly article
title: More on quadratic functions and maximal Artin-Schreier curves
zbMATH Open document ID: 1379.11066
author: Nurdagül Anbar
author: Wilfried Meidl
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2015-10-06T00:00:00Z
review text: Consider the Artin-Schreier cover of the \(\mathbb F_{p^n}\)-projective line given by \[ \mathcal{X}: y^p - y = \sum_{i=0}^{\lfloor n/2\rfloor} a_ix^{p^i+1} \] with \(a_i\in\mathbb F_{p^n}\). Let \(Q:\mathbb F_{p^n}\to\mathbb F_{p}\) be the function defined by \(Q(x) = \mathrm{Tr}_n\left(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1}\right)\), where \(\mathrm{Tr}_n(z)\)is the absolute trace of \(z\in\mathbb F_{p^n}\). The number of rational points of \(\mathcal X\) is\(N(\mathcal X) = 1+pN_0(Q)\) where \(N_0(Q)\) is the number of solutions of \(Q(x) = 0\). For an odd prime \(p\) and an even integer \(n\) with \(\gcd(n,p)>1\), the authors investigate quadratic functions \(Q\) of codimension \(k\) for which \(N(\mathcal X)\) attains the Hasse-Weil bound. They show that \(N_0(Q)\) is related to the Walsh coefficient of \(Q(x)\) at 0. Using this approach, they completely classify all minimal and maximal Artin-Schreier curves obtained from quadratic functions of codimension \(2\) with coefficients in \(\mathbb F_p\). This extends their work in [\textit{N. Anbar} and \textit{W. Meidl}, Finite Fields Appl. 30, 49--71 (2014; Zbl 1296.11074)] where they consider the case \(\gcd(n,p)=1\).
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 6489836
zbMATH Keywords: Artin-Schreier curve
zbMATH Keywords: partially bent function
zbMATH Keywords: quadratic function
zbMATH Keywords: Walsh transform
reviewed by: Ariane M. Masuda
MaRDI profile type: Publication
full work available at URL: s00200-015-0259-2
OpenAlex ID: W1995279938
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DOI: 10.1007/S00200-015-0259-2
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description: scientific article; zbMATH DE number 6489836