On The Reed-Muller Rule Under Channel Polarization

Being the first provably capacity-achieving codes, polar codes present a constructive solution to a long-standing open problem in information theory. To achieve this elusive goal, that is, capacity on arbitrary symmetric binary-input discrete memoryless channels, their construction explicitly takes into account the statistics of the target channel. Code construction may be modeled as selecting a row space from a certain basis of F-2(n). This is true for the much older class of Reed-Muller codes as well. In effect, the advent of polar codes reignited research on Reed-Muller codes. Recent results show that Reed-Muller codes are capacity-achieving on the binary erasure channel under maximum a-posteriori decoding. In this work, we assess how the row selections are related and conjecture that the Reed-Muller rule may provide a heuristic to select rows robust under successive cancellation decoding in the presence of varying channel conditions.