The Fractality of Polar and Reed-Muller Codes

The generator matrices of polar codes and Reed-Muller codes are obtained by selecting rows from the Kronecker product of a lower-triangular binary square matrix. For polar codes, the selection is based on the Bhattacharyya parameter of the row, which is closely related to the error probability of the corresponding input bit under sequential decoding. For Reed-Muller codes, the selection is based on the Hamming weight of the row. This work investigates the properties of the index sets pointing to those rows in the infinite blocklength limit. In particular, the Lebesgue measure, the Hausdorff dimension, and the self-similarity of these sets will be discussed. It is shown that these index sets have several properties that are common to fractals.