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

In general, the generator matrix sparsity is a critical factor in determining the encoding complexity of a linear code. Further, certain applications, e.g., distributed crowdsourcing schemes utilizing linear codes, require most or even all the columns of the generator matrix to have some degree of sparsity. 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 ∈ (0, 1], there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the 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 polar-based codes generated by the two schemes inherit several fundamental properties of polar codes with the original 2 × 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 β < 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 λ with λ ≈ 0.585, and with the error probability bounded by O (2 - N β ) 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 β < 0.5 with λ ≈ 0.631.