A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes

We present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distributions that arise in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, we can compute the entire weight distribution of certain polar cosets in time O(n^2). Any polar code can be represented as a disjoint union of such cosets; moreover, this representation extends to polar codes with dynamically frozen bits. This implies that our methods can be also used to compute the weight distribution of polar codes with CRC precoding, of polarization-adjusted convolutional (PAC) codes and, in fact, general linear codes. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To reduce the exponential complexity of our algorithm, we make use of the fact that polar codes have a large automorphism group, which includes the lower-triangular affine group LTA(m,2). We prove that LTA(m,2) acts transitively on certain sets of monomials, thereby drastically reducing the number of polar cosets we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of any polar code of length up to n=128

Computing the Weight Distribution of the Binary Reed-Muller Code R(4,9){\mathcal R} (4,9)

We compute the weight distribution of the ${\mathcal R} (4,9)$ by combining the approach described in D. V. Sarwate's Ph.D. thesis from 1973 with knowledge on the affine equivalence classification of Boolean functions. To solve this problem posed, e.g., in the MacWilliams and Sloane book [p. 447], we apply a refined approach based on the classification of Boolean quartic forms in $8$ variables due to Ph. Langevin and G. Leander, and recent results on the classification of the quotient space ${\mathcal R} (4,7)/{\mathcal R} (2,7)$ due to V. Gillot and Ph. Langevin