Fixed-threshold polar codes

We study a family of polar codes whose frozen set is such that it discards the bit channels for which the mutual information falls below a certain (fixed) threshold. We show that if the threshold, which might depend on the code length, is bounded appropriately, a coding theorem can be proved for the underlying polar code. We also give accurate closed-form upper and lower bounds to the minimum distance of the resulting code when the design channel is the binary erasure channel.