Capacity-achieving Polar-based LDGM Codes with Crowdsourcing Applications

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 ε> 2ε*, where ε8 = 1/6 - [5/3log4/3] ≈ 0.085, we show an explicit sequence of capacity-achieving codes with all the column weights of the generator matrix upper bounded by (log N)
^1+
ε
, where N is the code block length. The constructions are based on polar codes. Applications to crowdsourcing are also shown.