Rectangular Heffter arrays: a reduction theorem

Let m, n, s, k be four integers such that 3 < s < n, 3 < k < m and ms = nk. Set d = gcd(s, k). In this paper we show how one can construct a Heffter array H(m, n; s, k) starting from a square Heffter array H(nk/d; d) whose elements belong to d consecutive diagonals. As an example of application of this method, we prove that there exists an integer H(m, n; s, k) in each of the following cases: (i) d -0 (mod 4); (ii) 5 < d -1 (mod 4) and nk -3 (mod 4); (iii) d -2 (mod 4) and nk -0 (mod 4); (iv) d -3 (mod 4) and nk -0, 3 (mod 4). The same method can be applied also for signed magic arrays SMA(m, n; s, k) and for magic rectangles MR(m, n; s, k). In fact, we prove that there exists an SMA(m, n; s, k) when d > 2, and there exists an MR(m, n; s, k) when either d > 2 is even or d > 3 and nk are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when k is odd and s -0 (mod 4).(c) 2022 Elsevier B.V. All rights reserved.