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

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2006)

- Volume: 45, Issue: 1, page 31-34
- ISSN: 0231-9721

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topCaggegi, 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 -

## References

top- Beth T., Jungnickel, D, Lenz H.: Designs Theory., Bibliographisches Institut, Mannheim–Wien, 1985. MR0779284
- Caggegi A., Uniqueness of $A{G}_{3}(4,2)$, Italian Journal of Pure and Applied Mathematics 15 (2004), 9–16. Zbl1175.05028
- Hanani H., Balanced incomplete block designs and related designs, Discrete Math. 11 (1975), 255–369. (1975) Zbl0361.62067MR0382030
- Hughes D. R., Piper F. C.: Projective Planes., Springer-Verlag, Berlin–Heidelberg–New York, 1982, second printing. MR0333959

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