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Ovals in commutative semifield planes
Archiv der Mathematik, 1997A translation plane coordinatized by a semifield of odd order admits an orthogonal polarity whose absolute points form an oval \(\Omega\) (necessarily parabolic, having one point on the line at infinity). The collineation group of the plane which stabilizes such an \(\Omega\), acts 2-transitively on the affine points of \(\Omega\). In this paper, it is
ENEA, Maria Rosaria, KORCHMAROS, Gabor
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Baer involutions in semifield planes of even order
Geometriae Dedicata, 1974 MichaelJ. Ganley
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Fractional dimension of binary Knuth semifield planes
Journal of Combinatorial Designs, 2011AbstractWe prove that semifield planes π(𝕂2m) coordinatized by the commutative binary Knuth semifield 𝕂2m, m=nk (m odd) are fractional dimensional with respect to a subplane isomorphic to PG(2, 4) if either n=9 or n≡\0(mod 3) and one of the trinomials xn+xs+1, s∈{1, 2, 3, 5}, is irreducible over the Galois field 𝔽2. © 2012 Wiley Periodicals, Inc.
POLVERINO, Olga, Trombetti R.
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Transitive parabolic unitals in semifield planes
Journal of Geometry, 2006Every semifield plane with spread in PG(3,K), where K is a field admitting a quadratic extension K+, is shown to admit a transitive parabolic unital.
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A note on commutative semifield planes
Advances in Geometry, 2017Abstract Let q be an odd prime power. We prove that a planar function f from 𝔽 q to itself can be written as an affine Dembowski–Ostrom polynomial if and only if the projective plane derived from f is a commutative semifield plane.
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Net replacements in the Hughes-Kleinfeld semifield planes
Journal of Geometry, 2010If \(\alpha\) is an automorphism of a field \(K\), the cone \(C_\alpha\) in \(PG(3,k)\) consists of the points \(\{ (x_0,x_1,x_2,x_3) \) \( \;| \;x_0^\alpha x_1 = x_2^{\alpha + 1}\}\) with vertex \(v_0 = (0,0,0,1).\) A set of planes of \(PG(3,k)\) which partitions these points without \(v_0\) is a flock of \(C_\alpha\).
Cherowitzo, William E. +1 more
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A remark on symplectic semifield planes and Z 4-linear codes
Designs, Codes and Cryptography, 2012Kantor and Williams (Trans Am Soc 356:895---938, 2004) introduced a family of non-desarguesian symplectic semifields of even order and studied a number of structures connected with such semifields; namely, symplectic spreads, orthogonal spreads and Z 4-linear codes. Also, they provided equivalence results concerning such objects, although under certain
LUNARDON, GUGLIELMO +3 more
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More translation planes and semifields from Dembowski–Ostrom polynomials
Designs, Codes and Cryptography, 2012Let \(F=\mathrm{GF}(q^n)\), where \(q\) is an odd prime power and \(n\) is odd, and let \(\zeta \in \mathrm{GF}(q)\) be a non-square. In a recent paper, P. Müller and the author studied Dembowsky-Ostrom polynomials of the form \(P(X)=L(X)X\), with \(L(X)=\sum_{i=0}^{n-1} a_iX^{q^i}\), such that \((1)\) \(x \mapsto L(x)\) is bijective, \((2)\) \(|P(F^*)|
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On Elations of Derived Semifield Planes
Proceedings of the London Mathematical Society, 1977Johnson, N. L., Rahilly, Alan
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