Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this work, we propose an efficient algorithm to determine the number of minimum weight codewords of general PTPCs, which eliminates all redundant visits of nodes of the search tree, reducing the computational complexity from state-of-the-art algorithms typically by several orders of magnitude. This reduction in complexity allows, for the first time, the minimum distance properties to be directly considered in the code design of PTPCs. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Further, we design optimal convolutional polynomials for PAC codes with this algorithm, minimizing the number of minimum weight codewords.

翻译：预变换极化码是一类性能接近有限长容量界的编码。最小距离和最小重量码字数量是决定其性能的两个关键属性。本文提出一种高效算法，用于确定一般预变换极化码的最小重量码字数量。该算法消除了搜索树节点的所有冗余访问，相比现有最先进算法将计算复杂度通常降低数个数量级。这种复杂度降低使得首次能够在预变换极化码的编码设计中直接考虑最小距离特性。该算法在随机预变换里德-穆勒码和极化调整卷积码上得到验证。进一步地，我们利用该算法为极化调整卷积码设计最优卷积多项式，使最小重量码字数量最小化。