The Fractality of Polar and Reed-Muller Codes

The generator matrices of polar codes and ReedMuller codes are submatrices of a Kronecker product of a lower-triangular binary square matrix. These submatrices are chosen according to an index set pointing to rows, which for polar codes minimize the Bhattacharyya parameter, and which for Reed-Muller codes maximize the Hamming weight. This work investigates the properties of this index set 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 fulfill several properties which are common to fractals.