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Ovals in $${\mathbb{Z}}^2_{2p}$$. (English) Zbl 1453.51002
As a generalization of the classical definition for finite geometries, an oval in $$Z_{2p}^2$$, where $$p$$ is prime is defined as a set of $$2p+2$$ points with the property that no three of them are on a line, where a line is the set of $$(x,y)$$ satisfying $$ax+by+c =0$$ for some $$a, b, c$$, where $$a, b$$ and $$2p$$ are relatively prime. The author proves that these ovals only exist when $$p=3$$ or $$5$$ and that they are unique up to isomorphism.
##### MSC:
 5.1e+22 Blocking sets, ovals, $$k$$-arcs
##### Keywords:
arc; collinearity; ovals
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##### References:
 [1] R. C. Bose, Mathematical Theory of the Symmetrical Factorial Design Sankhya: The Indian Journal of Statistics (1933-1960) Vol. 8, No. 2 (Mar., 1947), 107-166 · Zbl 0038.09601 [2] Hirschfeld, JWP, Projective Geometries over Finite Fields (1998), Oxford: Clarendon Press, Oxford [3] Honold, T.; Landjev, I., On arcs in projective Hjelmslev planes, Discrete Math., 231, 1-3, 265-278 (2001) · Zbl 0994.51006 [4] Honold, T.; Landjev, I., On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic, Finite Fields Appl., 11, 2, 292-304 (2005) · Zbl 1069.51003 [5] J. Huizenga, The minimum size of complete caps in $$({\mathbb{Z}}/ n{\mathbb{Z}})^2$$, Electron. J. Combin. 13 (2006), no. 1, Research Paper 58, 19 pp. · Zbl 1165.51302 [6] Kleinfeld, E., Finite Hjelmslev planes, Illinois J. Math., 3, 403-407 (1959) · Zbl 0090.37106 [7] S. Kurz, Caps in $${\mathbb{Z}}_n^2$$, Serdica J. Computing 3 (2009), 159-178 · Zbl 1181.65086 [8] Kiermaier, M.; Koch, M.; Kurz, S., 2-arcs of maximal size in the affine and the projective Hjelmslev plane over $$Z_{25}$$, Adv. Math. Commun., 5, 2, 287-301 (2011) · Zbl 1252.94120 [9] Segre, B., Ovals in a finite projective plane, Canad. J. Math., 7, 414-416 (1955) · Zbl 0065.13402 [10] Z. Stȩpień, L. Szymaszkiewicz, Arcs in $${\mathbb{Z}}_{2p}^2$$, Journal of Combinatorial Optimization (2018), Volume 35, no. 2, 341-349. · Zbl 1386.05024
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