This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive asymptotic proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization [1], and without martingale convergence theorems for polarization process.
I. INTRODUCTIONArıkan [2] proposed binary polar codes as a class of provable symmetric capacity achieving codes with deterministic constructions and low encoding/decoding complexity for binary-input discrete memoryless channels (DMCs). In non-binary polar codes, there are two types of channel polarization: strong polarization [3], [10] and multilevel polarization [4]-[8]. Strong polarization asymptotically makes similar extremal channels to binary cases, i.e., either noiseless or pure noisy. On the other hand, multilevel polarization allows to converse several types of partially noiseless channels. It was independently shown in [3]-[8], [10] that both strong and multilevel channel polarization can achieve the symmetric capacity by showing rate of polarization for the Bhattacharyya parameters. Although the asymptotic distributions of strong polarization are fully and simply characterized by the symmetric capacity, the asymptotic distribution of multilevel channel polarization is, however, still an open problem.Recently, the authors [9] proposed a certain class of erasure channels together with the recursive formulas of the polar transforms for such a class. In addition, we [9] also clarified the asymptotic distribution of multilevel channel polarization for such a class when the input alphabet size q is a prime power. In this paper, we examine further the asymptotic distribution for general composite numbers q.