Some lattices from polynomial rings

We study some properties of the lattices defined as the kernel of the map $(a_1,\dots ,a_{q-1})\mapsto \prod (1-c_{i}x{)}^{a_i}\in ({\mathbb{F}}_q[x]\mathrm{/}(x^e){)}^{\ast }$, where ${\mathbb{F}}_q^{\ast }=\{c_1,\dots ,c_{q-1}\}$.