  For $q$ a power of a prime $p$, we study gamma factors of representations of
$GL_n(\mathbb{F}_q)$ over an algebraically closed field $k$ of positive
characteristic $\ell \neq p$. We show that the reduction mod $\ell$ of the
gamma factor defined in characteristic zero fails to satisfy the analogue of
the local converse theorem of Piatetski-Shapiro. To remedy this, we construct
gamma factors valued in arbitrary $\mathbb{Z}[1/p, \zeta_p]$-algebras $A$,
where $\zeta_p$ is a primitive $p$-th root of unity, for Whittaker-type
representations $\rho$ and $\pi$ of $GL_n(\mathbb{F}_q)$ and
$GL_m(\mathbb{F}_q)$ over $A$. We let $P(\pi)$ be the projective envelope of
$\pi$ and let $R(\pi)$ be its endomorphism ring and define new gamma factors
$\widetilde\gamma(\rho \times \pi) = \gamma((\rho\otimes_kR(\pi)) \times
P(\pi))$, which take values in the local Artinian $k$-algebra $R(\pi)$. We
prove a converse theorem for cuspidal representations using the new gamma
factors. When $n=2$ and $m=1$ we construct a different ``new'' gamma factor
$\gamma^{\ell}(\rho,\pi)$, which takes values in $k$ and satisfies a converse
theorem.
