Comparative Analysis of Polar Code Construction: BLER Bounds and Maximum Achievable Code Rate

Channel polarization is a process where a set of N communication channels is transformed into a set of polarized coordinate channels. These coordinate channels can be categorized into reliable and unreliable channels. The reliable channels can be used to transmit the information bits, while the unreliable ones are frozen, meaning they are fixed to a predetermined value (typically zero) and do not carry any information. This phenomenon is the foundation of the construction of polar codes. As the code length N grows, the coordinate channels become more polarized, leading to a situation where a coordinate channel $W_N^{(i)}$ becomes almost perfectly reliable, or extremely noisy. A polar code is defined by the triple $\left[ {N,K,{{\mathcal{A}}^c}} \right]$, where N is the block length, K is the number of information bits, and ${{\mathcal{A}}^c}$ is the frozen bit set. The ratio K/N denotes the code rate R. The Block Error Rate (BLER) is a key performance metric for polar codes, representing the probability that a block of codewords is decoded incorrectly.This study investigates the relationship between the BLER and the maximum achievable polar code rate R<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> across three channel models: the Binary Erasure Channel (BEC), the Binary Symmetric Channel (BSC), and the Additive White Gaussian Noise (AWGN) channel. Furthermore, we perform a comparative analysis of various approaches for constructing polar codes. Using MATLAB simulations, we compare the theoretical BLER upper bounds and maximum achievable code rates for each method, providing insights into the strengths and limitations of each approach. The results reveal that the BEC achieves the highest coding rate, followed by AWGN, while BSC yields the lowest coding rate for all methods. These findings provide insights into the design and optimization of polar code construction for diverse communication channels, offering guidance for their application.