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# Capacity-achieving Polar-based Codes with Sparsity Constraints on the Generator Matrices

Mar 2023

In this paper, we leverage polar codes and the well-established channelpolarization to design capacity-achieving codes with a certain constraint onthe weights of all the columns in the generator matrix (GM) while having alow-complexity decoding algorithm. We first show that given a binary-inputmemoryless symmetric (BMS) channel $W$ and a constant $s \in (0, 1]$, thereexists a polarization kernel such that the corresponding polar code iscapacity-achieving with the \textit{rate of polarization} $s/2$, and the GMcolumn weights being bounded from above by $N^s$. To improve the sparsityversus error rate trade-off, we devise a column-splitting algorithm and twocoding schemes for BEC and then for general BMS channels. The\textit{polar-based} codes generated by the two schemes inherit severalfundamental properties of polar codes with the original $2 \times 2$ kernelincluding the decay in error probability, decoding complexity, and thecapacity-achieving property. Furthermore, they demonstrate the additionalproperty 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 sequenceof capacity-achieving polar-based codes where all the GM column weights arebounded from above by $N^\lambda$ with $\lambda \approx 0.585$, and with theerror probability bounded by $O(2^{-N^{\beta}} )$ under a decoder withcomplexity $O(N\log N)$, is shown. The existence of similar capacity-achievingpolar-based codes with the same decoding complexity is shown for any BMSchannel and $\beta <0.5$ with $\lambda \approx 0.631$.

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