Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set

TL;DR

This paper explores the existence of special partitions and mappings in Abelian groups to construct magic rectangle sets with specific properties, advancing understanding in combinatorial design theory.

Contribution

It introduces new sufficient conditions for the existence of $ ext{MRS}_ ext{Gamma}(2k+1, 2^{ ext{alpha}}; c)$ based on zero-sum partitions via complete mappings.

Findings

Provided conditions for the existence of certain magic rectangle sets.

Connected group partitions to combinatorial designs.

Extended known results to new parameter cases.

Abstract

A complete mapping of a group $\Gamma$ is a bijection $\varphi\colon \Gamma\to \Gamma$ for which the mapping $x \mapsto x+\varphi(x)$ is a bijection. In this paper we consider the existence of a complete mapping $\varphi$ of $\Gamma$ and a partition $S_1,S_2,\ldots S_t$ of elements of $\Gamma$, such that $\sum_{s\in S_i}s=\sum_{s\in S_i}\varphi(s)=0$ for every $i$, $1 \leq i \leq t$. 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 $\Gamma$ of order $abc$, 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 $\delta \in \Gamma$. While a complete characterization of MRS$_\Gamma(a,b;c)$ exists for cases where $\{a,b\}\not=\{2k+1,2^{\alpha}\}$, the scenario where $\{a,b\}=\{2k+1,2^{\alpha}\}$ remains unsolved for $\alpha>1$. Using the partition of $\Gamma$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $\Gamma$-magic rectangle set MRS$_{\Gamma}(2k+1, 2^{\alpha};c)$ exists.