Signed magic arrays: existence and constructions

Let $m,n,s,k$ be four integers such that $1⩽s⩽n$, $1⩽k⩽m$ 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 $\mathrm{\Omega }\subset \mathbb{Z}$, where $\mathrm{\Omega }=\{0,\pm 1,\pm 2,\dots ,\pm (nk-1)\mathrm{/}2\}$ if $nk$ is odd and $\mathrm{\Omega }=\{\pm 1,\pm 2,\dots ,\pm nk\mathrm{/}2\}$ if $nk$ is even, satisfying the following requirements: $(a)$ every $\omega \in \mathrm{\Omega }$ 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 $s,k⩾5$ 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 $m,n,s,k$.