The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,λ)$ Designs with $v<100$

TL;DR

The paper solves the problem of classifying flag-transitive, point-imprimitive symmetric $2$-designs with $v<100$ by combining a theoretical construction based on the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $\mathrm{PSL}_{2}(7)$ with Camina–Zieschang decompositions that relate a design to its induced $\mathcal D_0$ and $\mathcal D_1$. It yields two new non-isomorphic $2$-$(64,28,12)$ designs with full automorphism group $G=2^{8}:\mathrm{PSL}_{2}(7)$, in addition to the known examples such as the complementary $\overline{PG_{5}(2)}$ and Kantor’s $\mathcal S^{-}(3)$. The results extend the existing classification for small $v$ and, together with prior work on flag-transitive, point-primitive designs, provide a complete picture for $v<100$. The work relies on both deep group-theoretic arguments and computational tools (GAP/DESIGN) to enumerate possibilities and verify isomorphism classes, culminating in a definitive catalog of such designs.

Abstract

A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,λ)$ designs with $v<100$ is provided. Apart from the known examples with $λ\leq 10$, the complementary design of $PG_{5}(2)$, and the $2$-design $\mathcal{S}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Braić-Golemac-Mandić-Vučičić \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$.

Theorems & Definitions (54)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Example 2.2
• proof
• Lemma 3.1
• Theorem 3.2: Camina-Zieschang
• Lemma 3.3
• proof