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 weights 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.


James Chin-Jen Pang

University of Michigan, Ann Arbor

Hessam Mahdavifar

University of Michigan

Sandeep Pradhan

University of Michigan
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Session Chair

Ido Tal

Technion - Israel Institute of Technology