An update on the existence of integer Heffter arrays

Abstract

An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $\Omega \subset \mathbb{Z}$ such that $\{\Omega ,-\Omega \}$ is a partition of the set $\{1,2,\dots ,2nk\}$ and such that the following conditions are satisfied: each row contains $s$ filled cells, each column contains $k$ filled cells, the elements in every row and column add up to $0$. It was conjectured by Dan Archdeacon that an integer $\stackrel{˝}{(}m,n;s;k)$ exists if and only if $ms=nk$, $3\le s\le n$, $3\le k\le m$ and $nk\equiv 0,3(\mathrm{mod}4)$. In this paper, we provide new constructions of these objects that allow us to prove the validity of Archdeacon's conjecture in each admissible case, except when $k=3,5$ and $s\not\equiv 0(\mathrm{mod}4)$ is such that $\gcd (s,k)=1$.