  In this work, we investigate guessing random additive noise decoding (GRAND)
with quantized soft input. First, we analyze the achievable rate of ordered
reliability bits GRAND (ORBGRAND), which uses the rank order of the reliability
as quantized soft information. We show that multi-line ORBGRAND can approach
capacity for any signal-to-noise ratio (SNR). We then introduce discretized
soft GRAND (DSGRAND), which uses information from a conventional quantizer.
Simulation results show that DSGRAND well approximates maximum-likelihood (ML)
decoding with a number of quantization bits that is in line with current soft
decoding implementations. For a (128,106) CRC-concatenated polar code, the
basic ORBGRAND is able to match or outperform CRC-aided successive cancellation
list (CA-SCL) decoding with codeword list size of 64 and 3 bits of quantized
soft information, while DSGRAND outperforms CA-SCL decoding with a list size of
128 codewords. Both ORBGRAND and DSGRAND exhibit approximately an order of
magnitude less average complexity and two orders of magnitude smaller memory
requirements than CA-SCL.
