Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.

翻译：极化码通常通过根据可靠性对合成比特信道进行排序来构建，这保证了其可达容量特性，但在短码长和中等码长下可能产生较差的低重量谱。近期的代数结果以闭式形式表达了单个比特信道对最小及近最小重量码字重数分布的贡献。在本工作中，我们将这些见解结合为一种混合（可靠性-权重）比特信道排序方法。我们定义了一种逐比特成本函数，其距离项源自最小重量码字的轨道枚举，并通过巴塔恰里亚型因子进行缩放，并证明所得混合构造在一类递减单项式码中最小化了截断的SC/ML联合界代理函数。通过剪枝/ML分解，我们将混合度量与SCL译码中的错误事件联系起来，并证明混合设计可作为基于可靠性构造的局部扰动，其渐近影响随码长趋近无穷大而消失。基于高斯近似和闭式重量贡献方法，在BPSK-AWGN信道下对短码长和中等码长的数值仿真结果，展示了纯可靠性构造与混合构造在最小距离、重数分布和联合界近似之间的权衡关系。所有证明均置于附录中。