  For any prime power $q$, Mori and Tanaka introduced a family of $q$-ary polar
codes based on $q$~by~$q$ Reed-Solomon polarization kernels. For transmission
over a $q$-ary erasure channel, they also derived a closed-form recursion for
the erasure probability of each effective channel. In this paper, we use that
expression to analyze the finite-length scaling of these codes on the $q$-ary
erasure channel with erasure probability $\epsilon\in(0,1)$. Our primary result
is that, for any $\gamma>0$ and $\delta>0$, there is a $q_{0}$ such that, for
all $q\geq q_{0}$, the fraction of effective channels with erasure rate at most
$N^{-\gamma}$ is at least $1-\epsilon-O(N^{-1/2+\delta})$, where $N=q^{n}$ is
the blocklength. Since this fraction cannot be larger than
$1-\epsilon-O(N^{-1/2})$, this establishes near-optimal finite-length scaling
for this family of codes. Our approach can be seen as an extension of a similar
analysis for binary polar codes by Hassani, Alishahi, and Urbanke.
  A similar analysis is also considered for $q$-ary polar codes with $m$ by $m$
polarizing matrices. This separates the effect of the alphabet size from the
effect of the matrix size. If the polarizing matrix at each stage is drawn
independently and uniformly from the set of invertible $m$ by $m$ matrices,
then the linear operator associated with the Lyapunov function analysis can be
written in closed form. To prove near-optimal scaling for polar codes with
fixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, we
conclude by stating two concrete mathematical conjectures that, if proven,
would imply near-optimal scaling for fixed~$q$.
