$2-(n^2,2n,2n-1)$ designs obtained from affine planes

Caggegi, Andrea. "$2-(n^2, 2n, 2n-1)$ designs obtained from affine planes." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 45.1 (2006): 31-34. <http://eudml.org/doc/32505>.

@article{Caggegi2006,
abstract = {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 $\{\mathcal \{D\}\}(\{\mathcal \{A\}\}, 2)$.},
author = {Caggegi, Andrea},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {$2-(n^2, 2n, 2n-1)$ designs; incidence structure; affine planes; affine plane; parallel classes; block design},
language = {eng},
number = {1},
pages = {31-34},
publisher = {Palacký University Olomouc},
title = {$2-(n^2, 2n, 2n-1)$ designs obtained from affine planes},
url = {http://eudml.org/doc/32505},
volume = {45},
year = {2006},
}

TY - JOUR
AU - Caggegi, Andrea
TI - $2-(n^2, 2n, 2n-1)$ designs obtained from affine planes
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2006
PB - Palacký University Olomouc
VL - 45
IS - 1
SP - 31
EP - 34
AB - 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 ${\mathcal {D}}({\mathcal {A}}, 2)$.
LA - eng
KW - $2-(n^2, 2n, 2n-1)$ designs; incidence structure; affine planes; affine plane; parallel classes; block design
UR - http://eudml.org/doc/32505
ER -