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，这种方法保证了容量可达性，但在短码和中长码下可能产生较差的小重量谱。最近的代数结果以闭式形式表达了单个比特信道对最小和近最小重量码字重数的贡献。在本工作中，我们将这些见解结合到一个混合（可靠性-权重）比特信道排序中。我们定义了一个每比特成本函数，其距离项源自最小重量码字的轨道枚举，并由一个Bhattacharyya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一个截断的SC/ML联合界代理。我们通过剪枝/ML分解将混合度量与SCL解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于无穷而消失。在BPSK-AWGN信道上，通过高斯近似和闭式重量贡献实现的短码和中长码数值结果，说明了纯基于可靠性的构造与混合构造在最小距离、重数和联合界近似方面的权衡。所有证明均置于附录中。