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 ǫ > 2ǫ * , where ǫ * = 1 6 − 5 3 log 4 3 ≈ 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+ǫ , where N is the code block length. The constructions are based on polar codes. Applications to crowdsourcing are also shown.
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