This study proposes modular arithmetic erasure channels (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Arıkan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also show that a recursive application of the polar transform for MAECs results in multilevel channel polarization, which is an asymptotic phenomenon that is characteristic of non-binary polar codes. Specifically, we establish a method to calculate the limiting proportions of the partially noiseless and noisy channels that are generated as a result of multilevel channel polarization for MAECs. In the particular case of MAECs, this calculation method solves an open problem posed by Nasser (2017) in the study of non-binary polar codes.

中文翻译：

---

模块化算术擦除通道及其多级通道极化

这项研究提出了模块化算术擦除通道 (MAEC)，这是一种新型的类擦除通道，其输入字母表不必是二进制的。此类包含二进制擦除通道 (BEC) 和一些其他已知的类擦除通道作为特殊情况。对于 MAEC，我们提供了类 Arıkan 极坐标变换的递归公式来模拟通道极化。换句话说，我们表明 MAEC 的合成通道与其他 MAEC 等效。这是著名的 BEC 极坐标变换递归公式的推广。使用我们的递归公式，我们还表明 MAEC 的极坐标变换的递归应用会导致多级信道极化，这是一种渐近现象，是非二进制极坐标码的特征。具体来说，我们建立了一种方法来计算由于 MAEC 的多级信道极化而产生的部分无噪声和有噪声信道的限制比例。在 MAEC 的特殊情况下，这种计算方法解决了 Nasser (2017) 在研究非二进制极性码时提出的一个开放问题。