engleski

Construction of 1-designs from group action

Let $G$ be a finite group acting on the set $\Omega$ in $k$ orbits, $\Omega_1$, $\Omega_2$, ... $\Omega_k$. From each orbit $\Omega_i, \ i=1, ..., k$, one can construct a transitive $1-$design with $|\Omega_i|$ points whose base block is $\cup_{;i=1};^s G_{;\alpha};.\beta_i, \ \alpha, \beta_1, ..., \beta_s \in \Omega_i$. Moreover, one can construct transitive $1-$design with $|\Omega_i|, \ i=1, ..., k$, points and $|\Omega_j|, \ j=1, ..., k$, blocks. Combining incidence matrices of those transitive $1-$designs we construct an incidence matrix of a non-transitive symmetric design with $\Omega$ as the point set. Using the describe method we construct a $1-$design on 416 points whose incidence matrix is an adjacency matrix of a strongly regular graph with parameters (416, 100, 36, 20) with the full automorphism group isomorphic to the group $G(2, 4):Z_2$.

1-design; group action

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