Signed magic arrays: existence and constructions

TL;DR

This paper investigates the existence and construction of signed magic arrays, providing necessary and sufficient conditions for their existence when certain parameters are even or odd coprime, expanding the understanding of these combinatorial structures.

Contribution

The paper introduces new constructions for signed magic arrays when n is even and s,k are odd coprime, establishing comprehensive existence conditions.

Findings

Constructed signed magic arrays for even n and odd coprime s,k

Established necessary and sufficient conditions for all admissible parameters

Extended the theory of combinatorial array designs

Abstract

Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $\Omega\subset \mathbb{Z}$, where $\Omega=\{0,\pm 1, \pm 2,\ldots, \pm (nk-1)/2\}$ if $nk$ is odd and $\Omega=\{\pm 1, \pm 2, \ldots, \pm nk/2\}$ if $nk$ is even, satisfying the following requirements: $(a)$ every $\omega \in \Omega$ appears once in the array; $(b)$ each row contains exactly $s$ filled cells and each column contains exactly $k$ filled cells; $(c)$ the sum of the elements in each row and in each column is $0$. In this paper we construct these arrays when $n$ is even and $s,k\geqslant 5$ are odd coprime integers. This allows us to give necessary and sufficient conditions for the existence of an $SMA(m,n; s,k)$ for all admissible values of $m,n,s,k$.