Self-converse Mendelsohn Designs with Odd Prime Block Size

Abstract:

      A Mendelsohn design $\MD(v, k, \lambda)$ is a pair $(X,{\cal B}),$ where $X$ is a $v$-set and ${\cal B}$ is a collection of $k$-tuples from $X$ such that each ordered pair from $X$ is contained in exactly $\lambda$ $k$-tuples of ${\cal B}$. An $\MD(v, k, \lambda)$ is called self-converse and denoted by $\SCMD(v, k, \lambda)=(X, {\cal B}, f)$, if there exists an isomorphic mapping $f$ from $(X, {\cal B})$ to $(X, {\cal B}^{-1})$. In this paper, using difference method, we give a constructive proof for the existence of $\SCMD(4mp,p,1),$ where $p$ is an odd prime and $m$ is a positive integer.