On the completeness of plane cubic curves over finite fields
type: Item
instance of: scholarly article
title: On the completeness of plane cubic curves over finite fields
zbMATH Open document ID: 1365.14041
author: Daniele Bartoli
author: Stefano Marcugini
author: Fernanda Pambianco
published in: Designs, Codes and Cryptography
publication date: 2017-04-13T00:00:00Z
review text: Let \(\mathbb K:= \mathrm{GF}(q)\) be the finite field of order \(q=p^a\), \(p\) prime. When the set \(\mathcal C=C(\mathbb K)\) of \(\mathbb K\)-rational points of a plane projective, absolutely irreducible, algebraic cubic curve \(C\) is a \textit{complete \((n,3)\)-arc} in \(\mathrm{PG}(2,q)\), the projective plane over the algebraic closure of \(\mathbb K\), where \(n=\# \mathcal C\)? This is the question addressed in this paper. Here, by Bézout's theorem, \(\mathcal C\) is a complete \((n,3)\)-arc in \(\mathrm{PG}(2,q)\) if and only if at least \(3\) points in \(\mathcal C\) are collinear, and if it is not contained in a \((n+1,3)-arc\). These objects are related to coding theory (specially with certain NMDS codes), and mathematical statistics; see e.g. \textit{J. W. P. Hirschfeld} and \textit{L. Storme}'s book [Dev. Math. 3, 201--246 (2001; Zbl 1025.51012)]. If \(C\) is nonsingular, \(q\geq 79\) with \(p>5\), then \(\mathcal C\) is in fact a complete \((n,3)\)-arc whenever its modular invariant is different from \(0\); otherwise, it admits a completion with at most \(n+3\) points [\textit{J. W. P. Hirschfeld} and \textit{J. F. Voloch}, J. Aust. Math. Soc., Ser. A 45, No. 2, 275--286 (1988; Zbl 0662.14018)]; this was improved by \textit{M. Giulietti} [Appl. Algebra Eng. Commun. Comput. 15, No. 1, 1--11 (2004; Zbl 1058.94019)]. In the paper under review the authors worked out the completeness for several classes of singular plane cubic curves. In particular, computer based results with MAGMA allow them to complete the aforementioned results on complete \((n,3)\)-arcs in \(\mathrm{PG}(2,q)\) for \(q\leq 81\)
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6705736
zbMATH Keywords: plane algebraic cubic
zbMATH Keywords: \((n,3)\)-arc
zbMATH Keywords: near MDS codes
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s10623-016-0215-6
OpenAlex ID: W2407511624
cites work: Small complete caps from singular cubics. II
cites work: Bounds on \((n,r)\)-arcs and their application to linear codes
cites work: The non-existence of some NMDS codes and the extremal sizes of complete \((n,3)\)-arcs in \(\mathrm{PG}(2,16)\)
cites work: Q4822436
cites work: The Magma algebra system. I: The user language
cites work: The complete (k, 3)-arcs of PG(2,q), q≤13
cites work: The number of points on an elliptic cubic curve over a finite field
cites work: Near-MDS codes over some small fields
cites work: On near-MDS codes
cites work: On plane arcs contained in cubic curves
cites work: On the extendibility of near-MDS elliptic codes
cites work: Q4370894
cites work: Q4550672
cites work: Q3812364
cites work: Simple constructions for elliptic cubic curves with specified small numbers of points
cites work: Nonsingular plane cubic curves over finite fields
cites work: Q3843291
cites work: Ovali ed altre curve nei piani di Galois di caratteristica due
cites work: Some results concerning \(\{(q+1)(n-1);\;n\}\)-arcs and \(\{(q+1)\;(n-1)+1;\;n\}\)-arcs in finite projective planes of order \(q\)
cites work: Abelian varieties over finite fields
DOI: 10.1007/S10623-016-0215-6
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description: scientific article; zbMATH DE number 6705736