On the completeness of plane cubic curves over finite fields
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 \(q\leq 81\)
scientific article; zbMATH DE number 6705736
scientific article