On the existence of integer relative Heffter arrays

Let v=2ms+t be a positive integer, where t divides 2ms, and let J be the subgroup of order t of the cyclic group Zv. An integer Heffter array Ht(m,n;s,k) over Zv relative to J is an m×n partially filled array with elements in Zv such that: (a) each row contains s filled cells and each column contains k filled cells; (b) for every x∈Zv∖J, either x or −x appears in the array; (c) the elements in every row and column, viewed as integers in [Formula presented], sum to 0 in Z. In this paper we study the existence of an integer Ht(m,n;s,k) when s and k are both even, proving the following results. Suppose that 4≤s≤n and 4≤k≤m are such that ms=nk. Let t be a divisor of 2ms. (a) If s,k≡0(mod4), there exists an integer Ht(m,n;s,k). (b) If s≡2(mod4) and k≡0(mod4), there exists an integer Ht(m,n;s,k) if and only if m is even. (c) If s≡0(mod4) and k≡2(mod4), then there exists an integer Ht(m,n;s,k) if and only if n is even. (d) Suppose that m and n are both even. If s,k≡2(mod4), then there exists an integer Ht(m,n;s,k).