Conference paper

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 kerneis. 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-Iength scaling of these codes on q-ary erasure channel with erasure probability is an element of is an element of(0,1). Our primary result is that, for any gamma > 0 and delta > 0, there is a q(0) such that, for all q >= q(0), the fraction of effective channels with erasure rate at most N-gamma is at least 1 - is an element of - O( N-1/2+delta), where N = q(n) is the blocklength. Since the gap to the channel capacity 1 - is an element of cannot vanish faster than O( N-1/2), this establishes near-optimal finite-Iength 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.


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