  In this paper, directional sequence entropy and directional Kronecker algebra
for $\mathbb{Z}^q$-systems are introduced. The relation between sequence
entropy and directional sequence entropy are established. Meanwhile,
direcitonal discrete spectrum systems and directional null systems are defined.
It is shown that a $\mathbb{Z}^q$-system has directional discrete spectrum if
and only if it is directional null. Moreover, it turns out that a
$\mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly
independent directions if and only if it has discrete spectrum.
