Some Additive $2-(v, 5,\lambda )$ Designs

Caggegi, Andrea

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 65-80.

Résumé

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.

Détail

Classification : 53B20, 53B30, 53C21
Keywords: Conformal mapping; geodesic mapping; conformal-geodesic mapping; initial conditions; (pseudo-) Riemannian space