Near-optimal finite-length scaling for polar codes over large alphabets
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 q-ary erasure channel with erasure probability ϵ ⋯ (0, 1). Our primary result is that, for any γ > 0 and δ > 0, there is a q0 such that, for all q ≥ q0, the fraction of effective channels with erasure rate at most N-γ is at least 1 - ϵ - O(N-1/2+δ), where N = qn is the blocklength. Since the gap to the channel capacity 1 - ϵ cannot vanish faster than 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 Mondelli, Hassani, and Urbanke.
