Classification problem for block-transitive 5-designs

Determine, up to isomorphism, all simple $5$ - $(v,k,\lambda)$ designs with $\lambda=2$ that admit a group of automorphisms acting transitively on their blocks; equivalently, classify all simple incidence structures $\mathcal{D}=(X,\mathcal{B})$ such that $|X|=v$ , every block has size $k$ , every $5$ -subset of $X$ lies in exactly two blocks, and some subgroup of $\operatorname{Aut}(\mathcal{D})$ is transitive on $\mathcal{B}$ .

References

Primary source

arXiv

Additional references

Block-Transitive 5-(v,k,2) Designs with k divides v — arXiv — Huijiao Hu, Zheng Huang, Shouqiang Shen

design-theory finite-group-theory

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 paper narrows the possibilities in an important special case, but the full classification remains open.

The problem asks for a classification of block-transitive $5$ -designs. The latest result treats a substantial parameter and group-action subcase rather than the general problem.

Known results

Affine-type automorphism groups are excluded in a relevant block-transitive Steiner $5$ -design subcase.
Flag-transitive Steiner $5$ -designs are classified: the Witt $5$ - $(12,6,1)$ and $5$ - $(24,8,1)$ designs are the only nontrivial cases in that narrower setting.

September 29, 2026 special-case classification

Huijiao Hu, Zheng Huang, and Shouqiang Shen claim that, under the Camina--Gagen condition and with $k$ dividing $v$ , affine-type groups are impossible and exactly two almost-simple possibilities remain: the $5$ - $(12,6,2)$ and $5$ - $(24,8,2)$ designs. This is a claimed advance, not an independently verified solution of the general problem.

Current status (as of September 2026): The stated special-case classification is claimed but unverified; the general classification of block-transitive $5$ -designs remains open.

Sources

No solutions have been posted yet.