Geodesic
Some Additive $2-(v, 5,\lambda )$ Designs
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 65-80

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Given a finite additive abelian group $G$ and an integer $k$, with $3\le k \le |G|$, denote by $\mathcal {D}_k (G)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C = \lbrace c_1, c_2,\dots , c_k\rbrace $ of $G$ such that $c_1 + c_2+\dots +c_k = 0$. It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that $\mathcal {D}_k (G)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block design submitted.] we know that $\mathcal {D}_3(G)$ is a 2-design if and only if $G$ is an elementary abelian 3-group. It is also known (see [Caggegi, A.: Some additive $2-(v,4,\lambda )$ designs Boll. Mat. Pura e Appl. 2 (2009), 1–3.]) that $G$ is necessarily an elementary abelian 2-group, if $\mathcal {D}_4(G)$ is a 2-design. Here we shall prove that $\mathcal {D}_5(G)$ is a 2-design if and only if $G$ is an elementary abelian 5-group.
Given a finite additive abelian group $G$ and an integer $k$, with $3\le k \le |G|$, denote by $\mathcal {D}_k (G)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C = \lbrace c_1, c_2,\dots , c_k\rbrace $ of $G$ such that $c_1 + c_2+\dots +c_k = 0$. It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that $\mathcal {D}_k (G)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block design submitted.] we know that $\mathcal {D}_3(G)$ is a 2-design if and only if $G$ is an elementary abelian 3-group. It is also known (see [Caggegi, A.: Some additive $2-(v,4,\lambda )$ designs Boll. Mat. Pura e Appl. 2 (2009), 1–3.]) that $G$ is necessarily an elementary abelian 2-group, if $\mathcal {D}_4(G)$ is a 2-design. Here we shall prove that $\mathcal {D}_5(G)$ is a 2-design if and only if $G$ is an elementary abelian 5-group.
Classification : 53B20, 53B30, 53C21
Keywords: Conformal mapping; geodesic mapping; conformal-geodesic mapping; initial conditions; (pseudo-) Riemannian space 
Caggegi, Andrea. Some Additive $2-(v, 5,\lambda )$ Designs. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 65-80. http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a4/
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[1] Beth, T., Jungnickel, D., Lenz, H.: Design Theory. 2nd ed., Cambridge University Press, Cambridge, 1999. | MR | Zbl

[2] Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group. arxiv 0806.3433v2, (2008), 1–8. | MR

[3] Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block design. submitted.

[4] Caggegi, A.: Some additive $2-(v,4,\lambda )$ designs. Boll. Mat. Pura e Appl. 2 (2009), 1–3. | Zbl

[5] Colbourn, C. J., Dinitz, J. H.: The CRC Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications, 2nd ed., Chapman & Hall/CRC Press, 2007. | MR | Zbl