On modular balanced partition designs

Abstract

Let $X$ be a finite set of integers with cardinality $\nu = \kappa \lambda$. 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 $\kappa$ blocks of size $\lambda$, such that every element of $X$ appears in exactly one block. If $\mathcal{A} = \{A_1, A_2, \ldots, A_{\kappa}\}$, then $\sum_{a\in A_i} a \equiv i \lambda \pmod{\nu}$, for each $i=1,2,\ldots,\kappa$ \item $\mathcal{B}$ is a partition of $X$ into $\lambda$ blocks of size $\kappa$, such that every element of $X$ appears in exactly one block. If $\mathcal{B} = \{B_1, B_2, \ldots, B_{\lambda}\}$, then $\sum_{b\in B_j} b \equiv j \kappa \pmod{\nu}$, for each $j=1,2,\ldots,\lambda$ \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.