Modular Arithmetic Erasure Channels and Their Multilevel Channel Polarization

Abstract

This study proposes a novel channel model called the modular arithmetic erasure channel, which is a general type of arbitrary input erasure-like channels containing the binary erasure channel (BEC) and some other previously-known erasure-like channels. For this channel model, we give recursive formulas of Arıkan-like polar transforms to simulate its channel polarization easily. In other words, similar to the polar transforms for BECs, we show that the synthetic channels of modular arithmetic erasure channels are again equivalent to the same channel models with certain transition probabilities, which can be easily calculated by explicit recursive formulas. We also show that Arıkan-like polar transforms for modular arithmetic erasure channels behave multilevel channel polarization, which is a phenomenon appeared in the study of non-binary polar codes; and thus, modular arithmetic erasure channels are informative toy problems of multilevel channel polarization. Furthermore, as a solution of an open problem in non-binary polar codes for special cases, we solve exactly and algorithmically the limiting proportions of partially noiseless synthetic channels, called the asymptotic distribution of multilevel channel polarization, for modular arithmetic erasure channels.

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

“…In practice, the asymptotic distribution is an important indicator of constructing polar codes [17]. As a special case, the authors of this paper [13], [14] solved the asymptotic distribution for non-binary-input erasure-like channel models proposed in [12]. Particularly, if the input alphabet size is not a prime power, then our previous works [13], [14] give a non-trivial method for calculating the exact asymptotic distribution algorithmically.…”

“…As a special case, the authors of this paper [13], [14] solved the asymptotic distribution for non-binary-input erasure-like channel models proposed in [12]. Particularly, if the input alphabet size is not a prime power, then our previous works [13], [14] give a non-trivial method for calculating the exact asymptotic distribution algorithmically. Recently, Nasser [5] characterized polarization levels, i.e., the support of the asymptotic distribution, for his proposed channels called automorphic-symmetric channels.…”

“…If G is a finite cyclic group, then the joint probability measure induced by (X, Y ) is equivalent to a modular arithmetic erasure channel [13, Definition 2] with uniform input distribution. As summarized in [13,, modular arithmetic erasure channels can be further reduced to many other erasure-like channels given in [15], [36]. In addition, if G is a finite elementary abelian group, then the joint probability measure induced by (X, Y ) is equivalent to a combination of linear channels [13, Definition 23] with uniform input distribution, where the linear channels are constructed on the set of subspaces of a finite-dimensional vector space over the prime field.…”

“…Proof of Lemma 5: See Appendix C. Since the positive divisors of a positive integer form a distributive lattice, Lemma 5 is a generalization of [13,Lemma 6] from a lattice of positive divisors with the stationary source setting to general distributive lattices with the nonstationary source setting. Therefore, as in [13, Lemma 7], we can obtain the following lemma.…”

“…In the following, we mention two examples of Theorem 2. 1) Modular Arithmetic Erasure Channels: As explained in Remarks 1 and 2, the erasure distribution defined in Definition 2 can be reduced to the modular arithmetic erasure channel [13,Definition 2], provided that G is a finite cyclic group. Since every cyclic group is locally cyclic, Theorem 2 can also be reduced to the authors' previous results [13], [14], which is described in a stationary setting.…”