1089-7798 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2019.2947910, IEEE Communications Letters JOURNAL OF LAT E X CLASS FILES, VOL. X, NO. Y, AUGUST 2015 1 Studying the binary erasure polarization subchannels using network reliability Vlad Dr˘ agoi and Valeriu Beiu, Member IEEE Abstract—In this article we analyze the reliability of the synthetic channels of a polar code designed for the binary erasure channel, and in particular the properties of their Bhattacharyya parameter. We apply techniques from network theory in order to establish fresh results. Firstly, we show how to efficiently approximate the Bhattacharyya parameter. Secondly, we identify asymptotically “good” channels for polar code. Thirdly, we determine a range of values for the erasure probability for which the selecting rule for the polar codes construction coincides with that of the Reed-Muller codes. Index Terms—Polar codes, Binary erasure channel, Reliability polynomials, Bhattacharyya parameter. I. I NTRODUCTION D ISCOVERED by Arikan [2], polar codes are one of the most promising code families as they attain the capacity of Binary Discrete Memoryless Channels (BDMC). That is why polar codes were proposed for the 5G technology [3]. One of the key ingredients for the construction of polar codes is the estimation of the reliability of the synthetic channels, W u , where u ∈{0, 1} m , and W is a BDMC. This can be done using several figures-of-merit, e.g., the mutual information or the Bhattacharyya parameter B. The message bits of a polar code are allocated to the k subchannels W u with the smallest B(W u ). When the length of the code n goes to infinity a polarization phenomenon takes place, i.e., W u become either “good” (noiseless) or “bad” (noisy) [2], and the fraction of good channels equals the capacity of the channel. In order to select “good” channels, several techniques were proposed [2], [4], [5]. Lately, a partial order over the set of synthetic channels was defined in [7], [8], and applied to the construction of polar codes in [6], [9], [10] . A special case discussed in [9] and [14] is that of polar codes over the binary erasure channel (BEC). In [14] the authors analyze the threshold behavior of B(W u ) and identify sufficient conditions to determine asymptotically “good” W u . The results of [14] are extended in [9], giving new sets of “good” synthetic channels for the BEC. In this letter we study B(W u ) over the BEC, modeled as the reliability polynomial (Rel) of particular two-terminal networks (2TNs). Here we will write B(W u )( p) in a Bernstein type basis, i.e., { p i (1 - p) 2 m -i ;0 ≤ i ≤ 2 m }, where p is the erasure probability. To any W u we associate a 2TN which leads to several applications. In [16] EXIT functions are also V. Dr˘ agoi and V. Beiu are with the Faculty of Exact Sciences of “Aurel Vlaicu” University of Arad, Arad, Romania, and V. Dr˘ agoi is also with University of Rouen Normandy, LITIS, Mont-Saint-Aignan, France. Manuscript received XXX; revised XXX. expressed in a similar fashion, however, the counting methods that we use here are different. Firstly, we will give a sharp estimate of B(W u ) in the two extreme cases, when p is close to 0 and when p is close to 1 (Theorem 3). Our approximation relies on the computation of the first non-zero coefficient of B(W u ), which requires O(| u |), where |·| is the Hamming weight. Our technique uses the Young diagrams associated to u ([8]), and takes into account not only | u | but also the position of the bits in u (unlike general bounds such as those in [1]), which makes it finer. This approach is constructive, i.e., it allows to determining which W u are “good” channels. Secondly, we will determine new synthetic channels that are asymptotically “good”, e.g., W (0 i 1 i 0 m-2i ) when i ≤ m/log m. Also, by analyzing B(W u )(1/2) we will extend the results from [14], [9] to a larger class of asymptotically “good” channels. Similarly to [14], the results we provide here can be used as a proxy for the construction of polar codes. Thirdly, we will analyze the conditions on p that enable the same selecting rule for both polar and Reed-Muller codes. Our results show that for almost all p ∈[0, 1] and m small the aforementioned condition is satisfied. Additionally, we will prove that for any m, when p is close to 0 or 1 the Reed- Muller code coincides with the polar code. This result is essentially described by the partial order given in Theorem 4, and shows the domination of the first/last term in the polarization behavior at low/high SNR regimes. While this is a known fact [15, Proposition 1], explicit thresholds were never mentioned. The results obtained here are valid for any DM code. Due to space limitations some proofs are just sketched. II. PRELIMINARIES In this article, m will denote a strictly positive integer, u a binary vector of length m, RM(r, m) and P the r th order Reed-Muller code, respectively a polar code. a) Polar codes over BEC: For any u = (u 1 ,..., u m )∈ {0, 1} m we define the synthetic channel W u = ((W u m ) ... ) u 1 as in [8]. When the initial channel is a BEC, B ( W (1) ) ( p) = 2p-p 2 and B ( W (0) ) ( p) = p 2 . For ordering the synthetic channels we use the partial order  from [8], which implies u  v ⇒ B(W u )( p)≤B(W v )( p), ∀p ∈[0, 1]. b) Two-terminal networks: Definition 1. Let n be a strictly positive integer. We say that N is a 2TN of size n if N is a circuit, made of n identical devices, that has two distinguished terminals: an input S, and an output T.