2024/01/25 by Carmen Amarra, Alice Devillers, Amarra, Carmen +3 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2401.13885
openalex publication_date 2024/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider 2-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on 2-designs which are block-transitive but not necessarily flag-transitive. In particular we use the concept of the ``array'' of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any s ≥ 2, there are infinitely many 2-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length s. Moreover an exhaustive computer search, using \sc Magma, seeking designs with e1e2e3 points (where each ei≤ 50) and a partition chain of length s=3, produced 57 such flag-transitive designs, among which only three designs arise from our construction -- so there is still much to learn.