×

Bilinear dual hyperovals from binary commutative presemifields. II. (English) Zbl 1375.05045

Summary: We construct a bilinear dual hyperoval \(S_c(S_1,S_2,S_3)\) from binary commutative presemifields \(S_1=(\mathrm{GF}(q),+,\circ)\) and \(S_2=(\mathrm{GF}(q),+, \ast)\), a binary presemifield \(S_3=(\mathrm{GF}(q),+,\star)\) which may not be commutative, and a non-zero element \(c\in \mathrm{GF}(q)\) which satisfies some conditions. We also determine the isomorphism problems under the conditions that \(S_1\) and \(S_2\) are not isotopic, and \(c\neq 1\). We also investigate farther on the isomorphism problem on the case that \(S_1\) and \(S_2\) are the Kantor commutative presemifields and \(S_3\) is the Albert presemifield.
For Part I, see [the author, ibid. 42, 93–101 (2016; Zbl 1348.05043)].

MSC:

05B25 Combinatorial aspects of finite geometries
51A45 Incidence structures embeddable into projective geometries
51E20 Combinatorial structures in finite projective spaces
51E21 Blocking sets, ovals, \(k\)-arcs

Citations:

Zbl 1348.05043
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, A. A., Generalized twisted fields, Pac. J. Math., 8, 1-8 (1961) · Zbl 0154.27203
[2] Biliotti, M.; Jha, V.; Johnson, N., The collineation groups of generalized twisted field planes, Geom. Dedic., 76, 97-126 (1999) · Zbl 0936.51003
[3] Coulter, R.; Henderson, M., Commutative presemifields and semifields, Adv. Math., 217, 282-304 (2008) · Zbl 1194.12007
[4] Dempwolff, U.; Edel, Y., Dimensional dual hyperovals and APN functions with translation groups, J. Comb., 39, 457-496 (2014) · Zbl 1292.05068
[5] Huybrechts, C.; Pasini, A., Flag transitive extensions of dual affine spaces, Beitr. Algebra Geom., 40, 503-532 (1999) · Zbl 0957.51004
[6] Kantor, W. M., Commutative semifields and symplectic spreads, J. Algebra, 270, 1, 96-114 (2003) · Zbl 1041.51002
[7] Kantor, W. M., Finite semifields, (Finite Geometries, Groups, and Computation, Proc. of Conf. at Pingree Park. Finite Geometries, Groups, and Computation, Proc. of Conf. at Pingree Park, CO, Sept. 2005 (2006), Walter de Gruyter: Walter de Gruyter Berlin-New York), 103-114 · Zbl 1102.51001
[8] Taniguchi, H., On the dual of the dual hyperoval from APN function \(f(x) = x^3 + t r(x^9)\), Finite Fields Appl., 15, 673-681 (2009) · Zbl 1185.51012
[9] Taniguchi, H., Bilinear dual hyperovals from binary commutative presemifields, Finite Fields Appl., 42, 93-101 (2016) · Zbl 1348.05043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.