Relaxed channel polarization for reduced complexity polar coding

Arıkan´s polar codes are proven to be capacity-achieving error correcting codes while having explicit constructions. They are characterized to have encoding and decoding complexities of l log l, for code length l. In this work, we construct another family of capacity-achieving codes that have even lower encoding and decoding complexities, by relaxing the channel polarizations for certain bit-channels. We consider schemes for relaxing the polarization of both sufficiently good and sufficiently bad bit-channels, in the process of channel polarization. We prove that, similar to conventional polar codes, relaxed polar codes also achieve the capacity of binary memoryless symmetric channels. We analyze the complexity reductions achievable by relaxed polarization for asymptotic and finite-length codes, both numerically and analytically. We show that relaxed polar codes can have better bit error probabilities than conventional polar codes, while having reduced encoding and decoding complexities.