Isomorphic designs that are not multiplier equivalent (Q1072558)

Two classes of isomorphic cyclic \(2\)-\((4^n,3,2)\) designs which are not multiplier equivalent are constructed. The question about the existence of isomorphic but multiplier inequivalent designs was formulated as an open problem in \textit{M. J. Colbourn} and \textit{R. A. Mathon}'s paper in [Ann. Discrete Math. 7, 215--253 (1980; Zbl 0438.05012)]. Some isomorphic \(2\)-\((v+1,3,2)\) designs invariant under the cyclic group \(Z_v\) but not equivalent under a multiplier of \(Z_v\) \((v=2.4^n)\) are also constructed.