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 × n partially filled array whose entries belong to the subset Ω ⊂ ℤ, where Ω = {0, ±1, ±2, . . . , ±(nk − 1)/2} if nk is odd and Ω = {±1, ±2, . . . , ±nk/2} if nk is even, satisfying the following requirements: (a) every ω ∈ Ω 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 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.