In this paper, we leverage polar codes and the well-established channel polarization to design capacity-achieving codes with a certain constraint on the weights of all the columns in the generator matrix (GM) while having a low-complexity decoding algorithm. We first show that given a binary-input memoryless symmetric (BMS) channel $W$ and a constant $s \in (0, 1]$, there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the \textit{rate of polarization} $s/2$, and the GM column weights being bounded from above by $N^s$. To improve the sparsity versus error rate trade-off, we devise a column-splitting algorithm and two coding schemes for BEC and then for general BMS channels. The \textit{polar-based} codes generated by the two schemes inherit several fundamental properties of polar codes with the original $2 \times 2$ kernel including the decay in error probability, decoding complexity, and the capacity-achieving property. Furthermore, they demonstrate the additional property that their GM column weights are bounded from above sublinearly in $N$, while the original polar codes have some column weights that are linear in $N$. In particular, for any BEC and $\beta <0.5$, the existence of a sequence of capacity-achieving polar-based codes where all the GM column weights are bounded from above by $N^\lambda$ with $\lambda \approx 0.585$, and with the error probability bounded by $O(2^{-N^{\beta}} )$ under a decoder with complexity $O(N\log N)$, is shown. The existence of similar capacity-achieving polar-based codes with the same decoding complexity is shown for any BMS channel and $\beta <0.5$ with $\lambda \approx 0.631$.

翻译：本文利用极化码与成熟的信道极化理论，在生成矩阵（GM）所有列权重受特定约束的条件下，设计具有低复杂度译码算法的容量可达编码。首先证明：对于给定二元输入无记忆对称（BMS）信道$W$及常数$s \in (0, 1]$，存在极化核使得对应极化码的极化速率达到$s/2$、GM列权重上界为$N^s$且容量可达。为优化稀疏性与误码率的权衡，针对BEC信道提出列分裂算法及两种编码方案，并推广至一般BMS信道。两种方案生成的基于极化码的编码继承了原始$2 \times 2$核极化码的若干基本性质，包括误码率衰减特性、译码复杂度及容量可达性。此外，其GM列权重上界在$N$中呈次线性增长，而原始极化码存在列权重随$N$线性增长的情况。特别地，对于任意BEC信道及$\beta <0.5$，存在容量可达的基于极化码的编码序列，其所有GM列权重上界为$N^\lambda$（$\lambda \approx 0.585$），且采用复杂度为$O(N\log N)$的译码器时误码率上界为$O(2^{-N^{\beta}})$。对任意BMS信道及$\beta <0.5$，在相同译码复杂度下可构造类似的容量可达编码，此时$\lambda \approx 0.631$。