Erasure Schemes Using Generalized Polar Codes: Zero-Undetected-Error Capacity and Performance Trade-offs

We study the performance of generalized polar (GP) codes when they are used for coding schemes involving erasure. GP codes are a family of codes which contains, among others, the standard polar codes of Arikan and Reed-Muller codes. We derive a closed formula for the zero-undetected-error capacity I_0^{GP}(W) of GP codes for a given binary memoryless symmetric (BMS) channel W under the low complexity successive cancellation decoder with erasure. We show that for every R < I_0^{GP}(W), there exists a generalized polar code of blocklength N and of rate at least R where the undetected-error probability is zero and the erasure probability is less than 2^{-N^{1/2-epsilon}}. On the other hand, for any GP code of rate I_0^{GP}(W) < R < I(W) and blocklength N, the undetected error probability cannot be made less than 2^{-N^{1/2+epsilon}} unless the erasure probability is close to 1.