Polar Codes with Successive Cancellation Decoding

Channel polarization transforms N identical channels into N channels with polarized capacities: some approach 1 (noiseless), others approach 0 (useless). Information bits on high-capacity channels are protected; low-capacity channels are frozen. As N grows, the fraction of high-capacity channels approaches the Shannon capacity. This structure allows simple successive cancellation decoding to achieve capacity.

Successive cancellation greedily decodes one bit at a time using previous bit estimates. If an early bit is decoded incorrectly, the error propagates to all downstream bits, causing a cascade of errors. This error floor arises from the dependence on past decisions. Belief propagation or list decoding mitigates this by exploring alternatives and deferring hard decisions.

Frozen bits are placed on channels with low estimated mutual information (low capacity). One method is to simulate the polarization process for the target SNR and blocklength, ranking channels by their capacity, then freezing the K lowest-capacity channels. This requires knowing SNR in advance. Adaptive polar codes can update the frozen bit set based on feedback or CSI estimation.

Instead of making a single hard decision at each step, list decoding maintains a list of candidate partial codewords and explores multiple paths. After processing all bits, the best path (by likelihood) is selected. This reduces error floor significantly but increases complexity to O(L N log N) where L is list size. List size 8-32 is typical; a good balance between performance and complexity.