ORBGRAND Is Exactly Capacity-achieving via Rank Companding

Among guessing random additive noise decoding (GRAND) algorithms, ordered reliability bits GRAND (ORBGRAND) has attracted considerable attention due to its efficient use of soft information and suitability for hardware implementation. It has also been shown that ORBGRAND achieves a rate very close to the capacity over additive white Gaussian noise channels with antipodal inputs. In this work, it is further established that, via suitably companding the ranks in ORBGRAND according to the inverse cumulative distribution function (CDF) of channel reliability, the resulting CDF-ORBGRAND algorithm exactly achieves the mutual information of general binary-input memoryless channels under symmetric input distribution, i.e., the symmetric capacity. This result is then applied to bit-interleaved coded modulation (BICM) systems to handle high-order input constellations. Via considering the effects of mismatched decoding due to both BICM and ORBGRAND, it is shown that CDF-ORBGRAND is capable of achieving the BICM capacity, which was initially derived by treating BICM as a set of independent parallel channels.