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

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 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. With this new perspective, we can fully utilize the property of the underlying channels and reduce the number of equivalence classes. The analysis is applied to binary symmetric channels and shows a great reduction of the number of equivalence classes compared with the original symmetry setting.