A magic rectangle set on Abelian groups

TL;DR

This paper characterizes the existence of group-based magic rectangle sets with uniform row and column sums, establishing conditions based on group properties and dimensions, expanding the theory of combinatorial designs.

Contribution

It provides necessary and sufficient conditions for the existence of $ ext{MRS}_ ext{Γ}(a, b; c)$ on Abelian groups, including new existence criteria and special cases.

Findings

Existence characterized for most parameter sets except specific cases.

Conditions depend on parity of dimensions and properties of the group $ ext{Γ}$.

Established criteria for the case when $c=1$, i.e., a single magic rectangle.

Abstract

A $\Gamma$-magic rectangle set $MRS_{\Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $\Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $\omega\in \Gamma$ and all column sums in every rectangle equal to a constant $\delta \in \Gamma$. In this paper we prove that for $\{a,b\}\neq\{2^{\alpha},2k+1\}$ where $\alpha$ and $k$ are some natural numbers, a $\Gamma$-magic rectangle set MRS$_{\Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|\Gamma|$ is odd or $\Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $\Gamma$-magic rectangle MRS$_{\Gamma}(a, b)$=MRS$_{\Gamma}(a, b;1)$.