ABSTRACT The largest prime that can be the order of an automorphism of a 2‐ design is , and all 2‐ designs with an automorphism of order 17 were classified by Tonchev. The symmetric 2‐ designs with automorphisms of an odd prime order were classified in Bouyukliev, Fack and Winne and Crnković and Rukavina. In this paper we give the classification of all symmetric 2‐ designs that admit an automorphism of order . It is shown that there are exactly nonisomorphic such designs. Furthermore, it is shown that the number of nonisomorphic 3‐ designs which have at least one derived 2‐ design with an automorphism of order 2, is .
This paper presents a significant advancement in the classification and construction of symmetric 2-(70, 24, 8) designs, which are fundamental objects in combinatorial design theory. These designs consist of 70 points and 70 blocks, where each block contains 24 points, and any pair of points is contained in exactly 8 blocks. The existence and classification of such designs are challenging problems.
The key innovation lies in a systematic computational approach to construct and classify these designs by analyzing the actions of a specific automorphism group. The authors focus on a cyclic automorphism group of order six, G = ⟨ρ⟩ ≅ Z₆ ≅ Z₂ × Z₃. A particularly fruitful action considered is one where the element ρ³ (of order two) fixes exactly 14 points (and blocks), and ρ² (of order three) fixes exactly 4 points (and blocks).
The method employed builds upon established techniques involving “orbit matrices.” For a design admitting a specific automorphism group, its points and blocks partition into orbits under the group action. The relationships between these orbits can be captured by an orbit matrix, from which incidence matrices representing actual designs can potentially be derived. This approach is especially powerful when the presumed automorphism group has a composite order, allowing for a decomposition into actions of subgroups. Specifically, the paper leverages Proposition 2.1, which describes how G-orbits of points/blocks relate to orbits under the action of subgroups of prime order (p, q for Z_pq).
Through extensive computation using custom programs and the MAGMA computer algebra system, the authors identify 3718 new non-isomorphic symmetric 2-(70, 24, 8) designs. This dramatically increases the number of previously known designs of these parameters. The constructed designs are further classified based on their full automorphism groups, revealing compositions like cyclic groups of order 6 (Z₆), groups of order 24 (A₄ × Z₂), Frobenius groups of order 42 (Frob₂₁ × Z₂), and groups of order 168 (E₈:Frob₂₁). The analysis also leads to a correction of a previous theorem regarding the classification of 2-(70, 24, 8) designs with a Frob₂₁ × Z₂ automorphism group. Furthermore, the paper investigates the binary codes associated with the incidence matrices of these designs, providing insights into their 2-rank properties.
The main prior ingredients for this work include the foundational principles of combinatorial design theory, particularly symmetric designs and their parameters (v, k, λ). Crucially, it relies on the theory of group actions on designs, including concepts of fixed points and orbits. Established results providing bounds on the number of fixed points for automorphisms (Propositions 2.2 and 2.3) are essential for narrowing down possible orbit distributions. The general “orbit matrix method” for constructing designs from presumed automorphism group actions, developed in earlier works, forms the algorithmic backbone. Finally, the ability to perform large-scale computational searches and classifications, facilitated by software like MAGMA, is indispensable.