Polar Codes: Robustness of the Successive Cancellation Decoder with Respect to Quantization

Polar codes provably achieve the capacity of a wide array of channels under successive decoding. This assumes infinite precision arithmetic. Given the successive nature of the decoding algorithm, one might worry about the sensitivity of the performance to the precision of the computation. We show that even very coarsely quantized decoding algorithms can lead to excellent performance. More concretely, we show that under successive decoding with an alphabet of cardinality only three, the decoder still has a threshold and this threshold is a sizable fraction of capacity. More generally, we show that if we are willing to transmit at a rate $\delta$ below capacity, then we need only $c \log(1/\delta)$ bits of precision, where $c$ is a universal constant.