Recognizing Recursive Designs from Their Incidence Structure
Overview
A design here consists of distinct, equally sized point subsets, called blocks, of size at least two; every point lies in the same positive number of blocks. For two recursive constructions, we recover hidden parts from point-block membership alone and determine all block-preserving point permutations, called automorphisms. This proves two conjectures of Amarra, Devillers and Praeger and classifies the exceptions. Integer relations among the membership vectors also recover the input of the single-partition construction without prior parameter information. Fixed finite construction recipes inherit the group and isomorphism results.
Original public questions
Amarra–Devillers–Praeger Conjectures 4.5 and 5.6: recursive designs
Carmen Amarra, Alice Devillers and Cheryl E. Praeger: Recursive constructions for block-transitive, poset-imprimitive two-designs — arXiv 2607.03029v1: Conjectures 4.5 and 5.6, determining the full point automorphism groups of Constructions 4.1 and 5.1 for the stated input 1-designs.
Original abstract (English)
Two recursive constructions of Amarra, Devillers and Praeger replace points by copies of a uniform design and produce natural wreath-product groups of automorphisms. We recover their defining partitions from the unlabelled incidence structures and determine the full automorphism groups, proving both of the authors' conjectures for arbitrary input 1-designs. The single-partition construction has exactly one exceptional family: two copies of a complete balanced bipartite graph. For the grid construction we also determine the full group in the two-row case, where independent permutations of the columns in each row enlarge the natural group. The proof uses the classical code generated by differences of block vectors. Its orthogonal space recovers fibers or grid columns; pair incidences handle the graph-input boundary. We compute the integer incidence quotients and all prime ranks of both constructions. The Smith normal form of the single-partition matrix gives parameter-free recovery of its input design. Fixed finite iterations inherit the group and isomorphism results.
Mathematical review
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