On modular balanced partition designs

Abstract

Let $X$ be a finite set of integers with cardinality $\hat{I}½=\hat{I}º\hat{I}»$. A \emph{modular balanced partition design} is a triplet $(X,\mathcal{A},\mathcal{B})$ satisfying the following conditions: \begin{itemize} \item $\mathcal{A}$ is a partition of $X$ into $\hat{I}º$ blocks of size $\hat{I}»$, such that every element of $X$ appears in exactly one block. If $\mathcal{A}=\{A_1,A_2,\dots ,A_{\hat{I}}º\}$, then ${\sum }_{a\in A_i}a\equiv i\hat{I}»(\mathrm{mod}\hat{I})½$, for each $i=1,2,\dots ,\hat{I}º$ \item $\mathcal{B}$ is a partition of $X$ into $\hat{I}»$ blocks of size $\hat{I}º$, such that every element of $X$ appears in exactly one block. If $\mathcal{B}=\{B_1,B_2,\dots ,B_{\hat{I}}»\}$, then ${\sum }_{b\in B_j}b\equiv j\hat{I}º(\mathrm{mod}\hat{I})½$, for each $j=1,2,\dots ,\hat{I}»$ \item $A_i\cap B_j$ has exactly one element, for any $A_i\in \mathcal{A}$, and $B_j\in \mathcal{B}$. \itemize} We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.