engleski

Some Menon designs having U(3, 3) as an automorphism group

There exists a unique symmetric (36, 15, 6) design D having $G'(2, 2) \cong U(3, 3)$ as an automorphism group. There is an incidence matrix M of D which is symmetric with 1 everywhere on the main diagonal, thus D admits a polarity for which all points are absolute. Therefore, M is an adjacency matrix of a strongly regular graph with parameters (36, 14, 4, 6). Using this design one can produce a series of symmetric designs with parameters $(4 \cdot (3 \cdot 2^k)^2, 2 \cdot (3 \cdot 2^k)^2 - 3 \cdot 2^k, (3 \cdot 2^k)^2 - 3 \cdot 2^k)$, $k \in N$, each of which admits an automorphism group isomorphic to the unitary group U(3, 3). There is an incidence matrix for each of these designs which is symmetric with constant diagonal. Therefore, these matrices correspond to adjacency matrices of strongly regular graphs.

symmetric design; Menon design; Hadamard matrix; strongly regular graph; automorphism group; unitary group.

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