On the Symmetry of Polar Codes for Symmetric Binary-Input Discrete Memoryless Channels

TL;DR

This paper analyzes the symmetry properties of polar codes on symmetric binary-input discrete memoryless channels, providing new theoretical insights into how output vectors can be grouped based on their transition probabilities.

Contribution

It introduces a novel analytical framework to characterize symmetries among received vectors in polar codes for B-DMC channels, extending prior work on their symmetry properties.

Findings

Theorems describing symmetry among received vectors

Equivalence classes of output vectors based on transition probabilities

Enhanced understanding of polar code structure for symmetric channels

Abstract

In this paper, we study the symmetry of polar codes on symmetric binary-input discrete memoryless channels (B-DMC). The symmetry property of polar codes is originally pointed out in Arikan's work for general B-DMC channels. With the symmetry, the output vector $y_1^N$ ($N$ be the block length) can be divided into equivalence classes in terms of their transition probabilities. In this paper, we present a new frame of analysis on the symmetry of polar codes for B-DMC channels. Theorems are provided to characterize the symmetries among the received vectors.