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For any prime power $q$, Mori and Tanaka introduced a family of $q$-ary polarcodes based on $q$~by~$q$ Reed-Solomon polarization kernels. For transmissionover a $q$-ary erasure channel, they also derived a closed-form recursion forthe erasure probability of each effective channel. In this paper, we use thatexpression to analyze the finite-length scaling of these codes on the $q$-aryerasure channel with erasure probability $\epsilon\in(0,1)$. Our primary resultis that, for any $\gamma>0$ and $\delta>0$, there is a $q_{0}$ such that, forall $q\geq q_{0}$, the fraction of effective channels with erasure rate at most$N^{-\gamma}$ is at least $1-\epsilon-O(N^{-1/2+\delta})$, where $N=q^{n}$ isthe blocklength. Since this fraction cannot be larger than$1-\epsilon-O(N^{-1/2})$, this establishes near-optimal finite-length scalingfor this family of codes. Our approach can be seen as an extension of a similaranalysis 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 theeffect of the matrix size. If the polarizing matrix at each stage is drawnindependently and uniformly from the set of invertible $m$ by $m$ matrices,then the linear operator associated with the Lyapunov function analysis can bewritten in closed form. To prove near-optimal scaling for polar codes withfixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, weconclude by stating two concrete mathematical conjectures that, if proven,would imply near-optimal scaling for fixed~$q$.

Near-Optimal Finite-Length Scaling for Polar Codes over Large Alphabets