一种计算极化码码重分布的确定性算法
A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes
摘要：
在这项工作中，我们提出了一个确定性的算法来计算极性码的整个重量分布。作为第一步，我们推导了一个有效的递归过程来计算沿任何解码路径的极化码的连续抵消解码中出现的权重分布。这解决了Polyanskaya，Davletshin和Polyanskii最近提出的公开问题。使用这个递归过程，在代码长度为n时，我们可以计算任何极陪集在时间O（n^2）内的权重分布。我们表明，任何极化码可以表示为这样的极性陪集的不相交的工会，此外，这种表示扩展到极化码与动态冻结位。然而，这种表示中的极陪集的数量与本文引入的参数呈指数关系，我们称之为混合因子。为了使极化码为递减单项式码时的算法复杂度上界，研究了其混合因子的取值范围。我们证明了在所有速率不超过1/2的递减单项码中，自对偶Reed-Muller码具有最大的混合因子。为了进一步降低算法的复杂度，我们利用了这样一个事实，即作为递减单项式码，极化码具有较大的自同构群。该自同构群包含块下三角仿射群（BLTA），而块下三角仿射群又包含下三角仿射群（LTA）。我们证明了一个子群的LTA行为传递的某些子集的减少单项码，从而大大减少了极陪集的数量，我们需要评估。这种复杂度降低使得可以计算长度n = 128的极化码的权重分布。
英文摘要：
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.
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