Abstract
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.
Highlights
The terminology and notation in this paper for designs and codes are as in [2, 3, 6].One of the main problems in design theory is that of classifying structures with given parameters
We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms
We describe the full automorphism groups of these designs and analyze their ternary codes
Summary
The terminology and notation in this paper for designs and codes are as in [2, 3, 6]. In this paper we manage to classify all symmetric (45,12,3) designs with nontrivial automorphisms. Kölmel [19] and Ćepulić [5] have independently constructed symmetric (45,12,3) designs having an automorphism of order 5. In this paper we give the classification of all symmetric (45,12,3) designs having a nontrivial automorphism group. The paper is organized as follows: after the brief introduction, in Section 2 we give basic information concerning the construction method, in Section 3 we describe the construction of symmetric (45,12,3) designs with nontrivial automorphisms and give a list of the designs and their full automorphism groups, Section 4 gives information about the codes of the constructed designs, and in Section 5 we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs. The codes have been analyzed using Magma [4]
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