Figure 1: Factor Graph In particular, consider a setup where (U1, U2) are two equiprobable bits that are encoded into (X1, X2) = (U1 ⊕ U2, U2). Then, (X1, X2) are mapped to (Y1, Y2) by two independent BMS channels with transition probabilities P(Y1 = y |X1 = x) = P(Y2 = y |X2 = x) = W (y|x). The factor graph for this setup is shown in Figure 1. Since the mapping from (U1, U2) to (X1, X2) is inv...

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent μ of polar codes for a memoryless channel qY |X with capacity I(qY |X) characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed ε ∈ (0, 1), the gap between the capacity I(qY |X) and the maximum non-asymptotic rate R n achi...

Polarization phenomenon over any finite field Fq with size q being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arıkan for the binary field, as well as its extension to a prime field by Şaşoğlu, Telatar, and Arıkan. In this paper, a necessary and sufficient condition of a matrix over a finite field Fq is shown under which...

So far in the course, we have seen three types of codes, summarized as follows: 1. Algebraic Codes • Work for worst-case errors. • Have polynomial time encoding and decoding algorithms 2. LDPCs • Work for worst-case errors as well, but a less many errors (smaller distance) compared to the Algebraic Codes. • Have linear time encoding and decoding algorithms and are simpler. 3. Polar Codes • Work...

We present the first family of binary codes that attains optimal scaling and quasi-linear complexity, at least for the binary erasure channel (BEC). In other words, for any fixed δ > 0, we provide codes that ensure reliable communication at rates within ε > 0 of the Shannon capacity with block length n = O(1/ε2+δ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n). Furth...

We discuss the performance of polar codes, the capacity-achieving channel codes, on wireless communication channel in this paper. By generalizing the definition of Bhattacharyya Parameter in discrete memoryless channel, we present the special expression of the parameter for Gaussian and Rayleigh fading the two continuous channels, including the recursive formulas and the initial values. We anal...