Relaxed Polar Codes

Polar codes are the latest breakthrough in coding theory, as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of discrete memoryless channels. Ar{\i}kan's polar encoder and successive cancellation decoder have complexities of $N \log N$, for code length $N$. Although, the complexity bound of $N \log N$ is asymptotically favorable, we report in this work methods to further reduce the encoding and decoding complexities of polar coding. The crux is to relax the polarization of certain bit-channels without performance degradation. We consider schemes for relaxing the polarization of both \emph{very good} and \emph{very bad} bit-channels, in the process of channel polarization. Relaxed polar codes are proved to preserve the capacity achieving property of polar codes. Analytical bounds on the asymptotic and finite-length complexity reduction attainable by relaxed polarization are derived. For binary erasure channels, we show that the computation complexity can be reduced by a factor of 6, while preserving the rate and error performance. We also show that relaxed polar codes can be decoded with significantly reduced latency. For AWGN channels with medium code lengths, we show that relaxed polar codes can have lower error probabilities than conventional polar codes, while having reduced encoding and decoding computation complexities.