Given the finite field \(\mathbb {F}_{q}\) , for a prime power q, in this paper we present a way of constructing spreads of \(\mathbb {F}_{q}^{n}\) . Specifically, through the field reduction technique, we will construct the well-known Desarguesian spread of \(\mathbb {F}_{q}^{n}\) , using for this purpose the action of an Abelian but not cyclic group. First, we construct a family of orbit codes of maximum distance using this group, and then we complete each of these codes to achieve the Desarguesian spread of the whole space, which will thus have a new orbital structure associated with an Abelian and non-cyclic group.
