  The classical Erd\H{o}s-Ginzburg-Ziv constant of a group $G$ denotes the
smallest positive integer $\ell$ such that any sequence $S$ of length at least
$\ell$ contains a zero-sum subsequence of length $\exp(G)$.
  In a recent paper, Caro and Schmitt generalized this concept, using the
$m$-th degree symmetric polynomial $e_m(S)$ instead of the sum of the elements
of $S$ and considering subsequences of a given length $t$. In particular, they
defined the higher degree Erd\H{o}s-Ginzburg-Ziv constants $EGZ(t,R,m)$ of a
finite commutative ring $R$ and presented several lower and upper bounds to
these constants.
  This paper aims to provide lower and upper bounds for $EGZ(t,R,m)$ in case
$R=\mathbb{F}_q^{n}$. The lower bounds here presented have been obtained,
respectively, using Lov\'asz Local Lemma and the Expurgation method and, for
sufficiently large $n$, they beat the lower bound provided by Caro and Schmitt
for the same kind of rings. Finally, we prove closed form upper bounds derived
from the Ellenberg-Gijswijt and Sauermann results for the cap-set problem
assuming that $q = p^k$, $t = p$, and $m=p-1$. Moreover, using the Slice Rank
method we derive a convex optimization problem that provides the best bounds
for $q = 3^k$, $t = 3$, $m=2$ and $k=2,3,4,5$.
