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ORBGRAND Is Exactly Capacity-achieving via Rank Companding

Zhuang Li, Wenyi Zhang

TL;DR

This work shows that a rank-companded variant of ORBGRAND, termed CDF-ORBGRAND, exactly achieves the symmetric capacity for binary-input memoryless channels by using the inverse CDF of LLR magnitudes to order error-pattern queries. The authors provide a rigorous information-theoretic analysis showing the achievable rate equals the mutual information, despite the decoder being a mismatched one. Extending the framework to bit-interleaved coded modulation (BICM), they prove that CDF-ORBGRAND achieves the BICM capacity, while standard ORBGRAND can incur rate losses in fading channels. The results offer a principled justification for rank-based decoders in high-order modulation, with practical implications for hardware-friendly decoding and future finite-length optimizations.

Abstract

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 of an additive white Gaussian noise channel under antipodal signaling. In this work, it is further established that, for general binary-input memoryless channels under symmetric input distribution, 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, 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 in the literature by treating BICM as a set of independent parallel channels.

ORBGRAND Is Exactly Capacity-achieving via Rank Companding

TL;DR

This work shows that a rank-companded variant of ORBGRAND, termed CDF-ORBGRAND, exactly achieves the symmetric capacity for binary-input memoryless channels by using the inverse CDF of LLR magnitudes to order error-pattern queries. The authors provide a rigorous information-theoretic analysis showing the achievable rate equals the mutual information, despite the decoder being a mismatched one. Extending the framework to bit-interleaved coded modulation (BICM), they prove that CDF-ORBGRAND achieves the BICM capacity, while standard ORBGRAND can incur rate losses in fading channels. The results offer a principled justification for rank-based decoders in high-order modulation, with practical implications for hardware-friendly decoding and future finite-length optimizations.

Abstract

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 of an additive white Gaussian noise channel under antipodal signaling. In this work, it is further established that, for general binary-input memoryless channels under symmetric input distribution, 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, 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 in the literature by treating BICM as a set of independent parallel channels.

Paper Structure

This paper contains 25 sections, 15 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Model
  4. GRAND and Its Variants
  5. Capacity-achieving Property of CDF-ORBGRAND
  6. Main Result
  7. Proof of Main Result
  8. Application of CDF-ORBGRAND in BICM
  9. BICM System Model
  10. BICM Capacity
  11. GRAND and Its Variants for BICM
  12. Rate Analysis
  13. Numerical Results
  14. Conclusion
  15. Proofs of Lemmas \ref{['lemma CDF-ORB exp']}, \ref{['lemma BICM CDF-ORB exp']}, and \ref{['lemma BICM ORB exp']}
  16. ...and 10 more sections

Key Result

Theorem 1

For the system model in Section subsec:system model, CDF-ORBGRAND achieves the symmetric capacity, i.e., in nats/channel use.

Figures (4)

  • Figure 1: Comparison between $I_{\text{ORB}}$ and $I_{\text{CDF-ORB}}$ under AWGN, AWLN, and Rayleigh fading channels. Since Rayleigh fading channels are described in complex baseband, the SNR is defined as the effective per-dimension SNR, corresponding to the in-phase component of the received signal.
  • Figure 2: Parallel channel model of BICM with ideal interleaving.
  • Figure 3: Gray (left) and SP (right) labelings for 16QAM. Here $0$ and $1$ correspond to $-1$ and $+1$, respectively.
  • Figure 4: Achievable rates of ORBGRAND, CDF-ORBGRAND, and BICM capacity for QPSK, 8PSK, and 16QAM over Rayleigh fading channel with perfect CSI. Line styles indicate curve type: solid for CDF-ORBGRAND and BICM capacity, dash-dot for ORBGRAND (\ref{['ORBGRAND rate BICM']}), and dashed for ORBGRAND under the assumption of ideal interleaving (\ref{['approx']}) in li2024orbgrand. Line colors indicate modulation format: black for QPSK, blue for 8PSK, red for 16QAM.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 6 more