Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms

Dean Crnkovic, Doris Dumicic Danilovic, Sanja Rukavina


We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms.
Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs.
We describe the full automorphism groups of these designs and analyze their ternary codes.
R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group,
which means that there are at least 5421 symmetric (45,12,3) designs.
Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs.
We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs
are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs
obtained from symmetric $(45,12,3)$ designs.

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ISSN: 2148-838X