  In this paper we study codes with sparse generator matrices. More
specifically, codes with a certain constraint on the weight of all the columns
in the generator matrix are considered. The end result is the following. For
any binary-input memoryless symmetric (BMS) channel and any epsilon > 2
epsilon*, where epsilon^* = \frac{1}{6}-\frac{5}{3}\log{\frac{4}{3}} \approx
0.085, we show an explicit sequence of capacity-achieving codes with all the
column wights of the generator matrix upper bounded by (\log N)^{1+epsilon},
where N is the code block length. The constructions are based on polar codes.
Applications to crowdsourcing are also shown.
