In this work, 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 distribution that arises 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, at code length n, we can compute the weight distribution of any polar cosets in time O(n^2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length n = 128.

翻译：本文提出了一种计算极化码完整重量分布的确定性算法。作为第一步，我们推导出一种高效递归过程，用于计算沿任意译码路径的极化码逐次抵消译码过程中出现的重量分布。这解决了Polyanskaya、Davletshin和Polyanskii近期提出的开放性问题。利用该递归过程，在码长为n时，可在O(n²)时间内计算任意极化陪集的重量分布。我们证明任意极化码可表示为此类极化陪集的不交并集；此外，该表示可推广至含动态冻结比特的极化码。然而，该表示中极化陪集的数量随本文引入的混合因子呈指数增长。为约束针对递减单项式码的算法复杂度，我们研究了其混合因子的取值范围。我们证明，在所有码率不超过1/2的递减单项式码中，自对偶Reed-Muller码具有最大的混合因子。为进一步降低算法复杂度，我们利用递减单项式码（即极化码）具有大自同构群这一性质。该自同构群包含块下三角仿射群，而后者又包含下三角仿射群。我们证明LTA的一个子群可传递作用于递减单项式码的特定子集上，从而大幅减少需要评估的极化陪集数量。这种复杂度降低使得计算码长n=128的极化码重量分布成为可能。