The Existence of Block-Transitive \(2-(q, 8, 1)\) Designs with \(q\) a Prime Power

Abstract

In this paper, we investigate the existence of \(2\)-\((v,8,1)\) designs admitting a block-transitive automorphism group \(G \leq \mathrm{ATL}(1,q)\). Using Weil’s theorem on character sums, the following theorem is proved:If a prime power \(q\) is large enough and \(q \equiv 57 \pmod{112}\), then there is always a \(2-(v,8,1)\) design which has a block-transitive, but non flag-transitive automorphism group \(G.\)