  In this paper we define a new class of partially filled arrays, called
relative Heffter arrays, that are a generalization of the Heffter arrays
introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where
$t$ divides $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$.
A $H_t(m,n; s,k)$ Heffter array over $\mathbb{Z}_v$ relative to $J$ is an
$m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that:
(a) each row contains $s$ filled cells and each column contains $k$ filled
cells; (b) for every $x\in \mathbb{Z}_v\setminus J$, either $x$ or $-x$ appears
in the array; (c) the elements in every row and column sum to $0$. Here we
study the existence of square integer (i.e. with entries chosen in
$\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\}$ and
where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$,
denoted by $H_k(n;k)$. In particular, we prove that for $3\leq k\leq n$, with
$k\neq 5$, there exists an integer $H_k(n;k)$ if and only if one of the
following holds: (a) $k$ is odd and $n\equiv 0,3\pmod 4$; (b) $k\equiv 2\pmod
4$ and $n$ is even; (c) $k\equiv 0\pmod 4$. Also, we show how these arrays give
rise to cyclic cycle decompositions of the complete multipartite graph.
