## Innovations in Incidence Geometry

### $j,k$-planes of order $4^3$

#### Abstract

A new class of translation planes of order $4 3$ is constructed and studied. These planes are a generalization of the $j$-planes discovered by Johnson, Pomareda and Wilke. These $j , k$-planes may be André replaced and the $j , k$-planes and the planes obtained by André replacement may be derived. There are thirteen new planes constructed and classified. Using ‘regular hyperbolic covers’, there are some new constructions of flat flocks of Segre varieties by Veronesians.

#### Article information

Source
Innov. Incidence Geom., Volume 2, Number 1 (2005), 1-34.

Dates
Accepted: 8 November 2005
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323257

Digital Object Identifier
doi:10.2140/iig.2005.2.1

Zentralblatt MATH identifier
1118.51005

#### Citation

Johnson, Norman L.; Vega, Oscar; Wilke, Fred W. $j,k$-planes of order $4^3$. Innov. Incidence Geom. 2 (2005), no. 1, 1--34. doi:10.2140/iig.2005.2.1. https://projecteuclid.org/euclid.iig/1551323257

#### References

• L. Bader, A. Cossidente and G. Lunardon, Generalizing flocks of $Q^{+}(3,q)$, Adv. Geom. 1 (2001), 323–331.
• ––––, On flat flocks, preprint.
• S. Ball, J. Bamberg, M. Lavrauw and T. Penttila, Symplectic spreads, Des. Codes Cryptogr. 32 (2004), no. 1–3, 9–14.
• C. Culbert and G. L. Ebert, Circle geometry and three-dimensional subregular translation planes, Innovations in Incidence Geometry 1 (2005), no. 1, 3–18.
• D. A. Foulser, Collineation groups of generalized André planes, Canad. J. Math. 21 (1969), 358–369.
• Y. Hiramine, V. Jha and N. L. Johnson, Cubic extensions of flag-transitive planes, I: Even order, Int. J. Math. Math. Sci. 25 (2001), 533–547.
• J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, New York, 1991.
• V. Jha and N. L. Johnson, Derivable nets defined by central collineations, J. Combin. Inform. System Sci. 11 (1986), no. 2–4, 83–91.
• ––––, André flat flocks, Note di Matematica (to appear).
• ––––, Semifield flat flocks, Adv. Geom. (to appear).
• N. L. Johnson, A note on net replacement in transposed spreads, Canad. Math. Bull. 28 (1985), no. 4, 469–471.
• ––––, Homology groups of translation planes and flocks of quadratic cones, I: The structure, Bull. Belg. Math. Soc. (to appear).
• ––––, Spreads in $PG(3,q)$ admitting several homology groups of order $q+1$, Note di Matematica (to appear).
• N. L. Johnson and T. G. Ostrom, Inherited groups and kernels of derived translation planes, European J. Combin. 11 (1990), 145–149.
• N. L. Johnson and R. Pomareda, Translation planes with many homologies, II: Odd order, J. Geom. 72 (2001), 77–107.
• N. L. Johnson, R. Pomareda and F. W. Wilke, $j$-planes, J. Combin. Theory Ser. A 56 (1991), no. 2, 271–284.
• N. L. Johnson and O. Vega, Symplectic Spreads and Symplectically Paired Spreads, preprint.
• W. M. Kantor, Orthogonal spreads and translation planes, Progress in Algebraic Combinatorics (Fukuoka, 1993), Advanced Studies in Pure Mathematics, vol. 24, Mathematical Society of Japan, Tokyo, 1996, pp. 227–242.
• ––––, Isomorphisms of symplectic spreads, European J. Combin. (to appear).
• H. Lüneburg, Über die Anzahl der Dickson'schen Fastkörper gegebener Ordnung, Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Perugia, 1970), Ist. Mat. Univ. Perugia, Perugia, 1971, pp. 319–322.
• ––––, Translation Planes, Springer-Verlag, Berlin–New York, 1980.
• T. G. Ostrom, Linear transformations and collineations of translation planes, J. Algebra 14 (1970), 405–416.
• R. Pomareda, Hyper-reguli in projective space of dimension $5$, Mostly Finite Geometries (Iowa City, 1996), Lecture Notes in Pure and Appl. Math., vol. 190, Dekker, New York, 1997, pp. 379–381.
• M. E. Williams, $Z_{4}$-linear Kerdock codes, orthogonal geometries, and non-associative division algebras, Ph.D. thesis, University of Oregon, 1995.