Signed magic arrays: existence and constructions

Let m,n,s,kbe four integers such that 1⩽s⩽n, 1⩽k⩽mand ms=nk. A signed magic array SMA(m,n;s,k)is an m×npartially fifilled array whose entries belong to the subset Ω⊂ℤ, where Ω={0,±1,±2,...,±(nk−1)/2}if nkis odd and Ω={±1,±2,...,±nk/2}if nkis even, satisfying the following requirements: (a)every ω∈Ωappears once in the array; (b)each row contains exactly sfifilled cells and each column contains exactly kfifilled cells; (c)the sum of the elements in each row and in each column is 0. In this paper we construct these arrays when nis even and s,k⩾5are coprime integers. This allows us to provide a complete answer to a problem posed in 2017 by Khodkar, Schulz and Wagner, giving the necessary and sufficient conditions for the existence of an SMA(m,n;s,k)for all admissible values of m,n,s,k.