In this letter, we introduce a class of discrete memoryless broadcast channels, called noisy Blackwell channels, which generalize the Blackwell channel to include noise and to more than two users. We design the polar code for the noisy Blackwell channel based on Marton's coding scheme. We choose auxiliary random variables and the mapping from auxiliary random variables to the channel input such that the sum-rate capacity can be achieved very closely. Furthermore, we show that the designed polar code can be applied to multi-level cell flash memories for improving random I/O performance. The simulation results demonstrate that our designed polar codes achieve good performance even for a large number of users.

We consider a discrete memoryless broadcast relay network that consists of one transmitter, two receivers and a relay. The transmitter sends independent messages to each receiver using Marton coding. The relay performs decode-andforward using another Marton coding. We show the achievable rate region of the proposed scheme and also provide an outer bound for the channel.