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In this paper, we study codes with sparse generator matrices. More specifically, low-density generator matrix (LDGM) codes with a certain constraint on the weight of the columns in the generator matrix are considered. In this paper, it is first shown that when a BMS channel W and a constant s>0 are given, there exists a polarization kernel such that the corresponding polar code is capacity-achieving and the column weights of the generator matrix (GM) are bounded from above by $N^s$. Then, a general construction based on a concatenation of polar codes and a rate-$1$ code, and a new column-splitting algorithm that guarantees a much sparser GM, is given. More specifically, for any BMS channel and any $ε> 2ε^*$, where $ε^* \approx 0.085$, an existence of a sequence of capacity-achieving codes with all the GM column weights upper bounded by $(\log N)^{1+ε}$ is shown. Furthermore, two coding schemes for BEC and BMS channels, based on a second column-splitting algorithm, are devised with low-complexity decoding that uses successive-cancellation. The second splitting algorithm allows for the use of a low-complexity decoder by preserving the reliability of the bit-channels observed by the source bits, and by increasing the code block length. The concatenation-based construction can also be applied to the random linear code ensemble to yield capacity-achieving codes with all the GM column weights being $O(\log N)$ and with (large-degree) polynomial decoding complexity.