| Title:
|
$2-(n^2, 2n, 2n-1)$ designs obtained from affine planes (English) |
| Author:
|
Caggegi, Andrea |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
31-34 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The simple incidence structure ${\mathcal D}({\mathcal A}, 2)$ formed by points and unordered pairs of distinct parallel lines of a finite affine plane ${\mathcal A} = ({\mathcal P}, {\mathcal L})$ of order $n>2$ is a $2-(n^2,2n,2n-1)$ design. If $n = 3$, ${\mathcal D}({\mathcal A}, 2)$ is the complementary design of ${\mathcal A}$. If $n = 4$, ${\mathcal D}({\mathcal A}, 2)$ is isomorphic to the geometric design $AG_3(4, 2)$ (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a $2-(n^2,2n,2n-1)$ design to be of the form ${\mathcal D}({\mathcal A}, 2)$ for some finite affine plane ${\mathcal A}$ of order $n>4$. As a consequence we obtain a characterization of small designs ${\mathcal D}({\mathcal A}, 2)$. (English) |
| Keyword:
|
$2-(n^2, 2n, 2n-1)$ designs |
| Keyword:
|
incidence structure |
| Keyword:
|
affine planes |
| MSC:
|
05B05 |
| MSC:
|
05B25 |
| MSC:
|
51E15 |
| idZBL:
|
Zbl 1125.05015 |
| idMR:
|
MR2321294 |
| . |
| Date available:
|
2009-08-21T07:04:29Z |
| Last updated:
|
2012-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133442 |
| . |
| Reference:
|
[1] Beth T., Jungnickel, D, Lenz H.: Designs Theory. : Bibliographisches Institut, Mannheim–Wien.1985. MR 0779284 |
| Reference:
|
[2] Caggegi A.: Uniqueness of $AG_3(4, 2)$.Italian Journal of Pure and Applied Mathematics 15 (2004), 9–16. Zbl 1175.05028 |
| Reference:
|
[3] Hanani H.: Balanced incomplete block designs and related designs.Discrete Math. 11 (1975), 255–369. Zbl 0361.62067, MR 0382030 |
| Reference:
|
[4] Hughes D. R., Piper F. C.: Projective Planes. : Springer-Verlag, Berlin–Heidelberg–New York.1982, second printing. MR 0333959 |
| . |
