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# André embeddings of affine planes

Joseph Thas (UGent) and Hendrik Van Maldeghem (UGent)
(2010) 523. p.123-131
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Abstract
An Andre embedding is a representation of a point-line geometry S with approximately s(2) points on a line in a planar space with approximately s points per line, but such that the lines of S are contained in planes of the planar space. An example is the Andre representation (also sometimes called the Bose-Bruck representation) of an affine translation plane of order q(2) (with kernel of order at least q) in 4-dimensional affine space AG(4, q), using a line spread at infinity. In this paper, we classify all Andre embeddings of affine planes of order q(2) in PG(4, q), q > 2, and obtain, besides the natural extension to PG(4, q) of the above example, two other related constructions. We also consider Andre embeddings of affine planes of order q(2) in PG(d, q), with d > 4 and q > 2.

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## Citation

Chicago
Thas, Joseph, and Hendrik Van Maldeghem. 2010. “André Embeddings of Affine Planes.” In Contemporary Mathematics, ed. AA Bruen and DL Wehlau, 523:123–131. Providence, RI, USA: American Mathematical Society.
APA
Thas, J., & Van Maldeghem, H. (2010). André embeddings of affine planes. In A. Bruen & D. Wehlau (Eds.), Contemporary Mathematics (Vol. 523, pp. 123–131). Presented at the Conference on Error-Control Codes, Information Theory and Applied Cryptography, Providence, RI, USA: American Mathematical Society.
Vancouver
1.
Thas J, Van Maldeghem H. André embeddings of affine planes. In: Bruen A, Wehlau D, editors. Contemporary Mathematics. Providence, RI, USA: American Mathematical Society; 2010. p. 123–31.
MLA
Thas, Joseph, and Hendrik Van Maldeghem. “André Embeddings of Affine Planes.” Contemporary Mathematics. Ed. AA Bruen & DL Wehlau. Vol. 523. Providence, RI, USA: American Mathematical Society, 2010. 123–131. Print.
```@inproceedings{1199279,
abstract     = {An Andre embedding is a representation of a point-line geometry S with approximately s(2) points on a line in a planar space with approximately s points per line, but such that the lines of S are contained in planes of the planar space. An example is the Andre representation (also sometimes called the Bose-Bruck representation) of an affine translation plane of order q(2) (with kernel of order at least q) in 4-dimensional affine space AG(4, q), using a line spread at infinity. In this paper, we classify all Andre embeddings of affine planes of order q(2) in PG(4, q), q {\textrangle} 2, and obtain, besides the natural extension to PG(4, q) of the above example, two other related constructions. We also consider Andre embeddings of affine planes of order q(2) in PG(d, q), with d {\textrangle} 4 and q {\textrangle} 2.},
author       = {Thas, Joseph and Van Maldeghem, Hendrik},
booktitle    = {Contemporary Mathematics},
editor       = {Bruen, AA and Wehlau, DL},
isbn         = {9780821849569},
issn         = {0271-4132},
language     = {eng},