© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2310.10737 · IEEE Transactions on Wireless Communications

Soft-Output (SO) GRAND and Iterative Decoding to Outperform LDPC Codes

Peihong Yuan, Muriel Médard, Kevin Galligan, Ken R. Duffy

Published Jan 24, 2025• 27 citations

Abstract

We establish that a large, flexible class of long, high redundancy error correcting codes can be efficiently and accurately decoded with guessing random additive noise decoding (GRAND). Performance evaluation demonstrates that it is possible to construct simple product codes with lengths of approximately 200 to 4000 bits and rates between 0.2 and 0.8 that outperform low-density parity-check (LDPC) codes from the 5G New Radio standard in both AWGN and fading channels. The concatenated structure enables many desirable features, including: low-complexity hardware-friendly encoding and decoding; significant flexibility in length and rate through modularity; and high levels of parallelism in encoding and decoding that enable low latency. Central is the development of a method through which any soft-input (SI) GRAND algorithm can provide soft-output (SO) in the form of an accurate a-posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that each element of the list is correct. The distinguishing feature of soft-output GRAND (SOGRAND) is the provision of an estimate that the correct decoding has not been found, even when providing a single decoding. Per-block SO can be converted into accurate per-bit SO by a weighted sum that includes a term for the SI. Implementing SOGRAND adds negligible computation and memory to the existing decoding process, and using it results in a practical, low-latency alternative to LDPC codes.

DOI: 10.1109/twc.2025.3530880

Related papers

SOGRAND decoding of LDPC codes

Ken R. Duffy, Jiewei Feng, Lukas Rapp +1 more

Long forward error correction codes are typically constructed by concatenating shorter component codes that are then decoded through iterative Soft-Input Soft-Output (SISO) of their components. The recently introduced Soft Output Guessing Random Additive Noise Decoding (SOGRAND) has been shown to enable accurate SISO component decoding for a broad range of component codes. Here we establish that by specializing its SISO computation to Single Parity Check codes, SOGRAND offers an alternative existing Check Node (CN) update for decoding Low Density Parity Check codes. Simulation results demonstrate similar or better decoding performance than Gallager's sum-product algorithm and norm-min-sum, while offering two distinct low complexity, hardware friendly CN update algorithms.

arXiv:2607.04045•Jul 4, 2026•0 citations

Ordered Reliability Bits Guessing Random Additive Noise Decoding

Ken R. Duffy, Wei An, Muriel Médard

Error correction techniques traditionally focus on the co-design of restricted code-structures in tandem with code-specific decoders that are computationally efficient when decoding long codes in hardware. Modern applications are, however, driving demand for ultra-reliable low-latency communications (URLLC), rekindling interest in the performance of shorter, higher-rate error correcting codes, and raising the possibility of revisiting universal, code-agnostic decoders. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can accurately decode any moderate redundancy block-code. It is designed with efficient circuit implementation in mind, and determines accurate decodings while retaining the original hard detection GRAND algorithm's suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide excellent soft decision block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with modest complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder. ORBGRAND offers the possibility of an accurate, energy efficient soft detection decoder suitable for delivering URLLC in a single hardware realization.

arXiv:2202.13951•IEEE Transactions on Signal Processing•Jan 1, 2022•133 citations

Soft Maximum Likelihood Decoding using GRAND

Amit Solomon, Ken R. Duffy, Muriel Médard

Maximum Likelihood (ML) decoding of forward error correction codes is known to be optimally accurate, but is not used in practice as it proves too challenging to efficiently implement. Here we propose a development of a previously described hard detection ML decoder called Guessing Random Additive Noise Decoding (GRAND). We introduce Soft GRAND (SGRAND), a ML decoder that fully avails of soft detection information and is suitable for use with any arbitrary high-rate, short-length block code. We assess SGRAND's performance on Cyclic Redundancy Check (CRC)-aided Polar (CA-Polar) codes, which will be used for all control channel communication in 5G New Radio (NR), comparing its accuracy with CRC-Aided Successive Cancellation List decoding (CA-SCL), a state-of-the-art soft-information decoder specific to CA-Polar codes.

arXiv:2001.03089•International Conference on Communications•Jun 1, 2020•119 citations

Efficient Soft-Output Guessing for Enhanced Quantum Tanner Code Decoding

Lukas Rapp, Muriel Médard, Eugene Tang +1 more

We introduce a generalized low-density parity-check decoding framework for quantum Tanner codes utilizing soft-output guessing random additive noise decoding (SOGRAND). By soft-output decoding entire component codes rather than individual parity checks, we mitigate the effects of trapping sets and cycles, resulting in improved convergence. Because our decoder preserves the message passing structure of standard belief propagation (BP), it is compatible with many BP enhancements. We demonstrate this with OSD postprocessing, quaternary BP, and RelayBP, each yielding further gains. Standalone SOGRAND outperforms the standard BP+OSD baseline by over two orders of magnitude in logical error rate. In combination with the Relay principle, SOGRAND outperforms RelayBP by over one order of magnitude, providing a way forward for scalable decoding of the emerging class of Tanner-code-based quantum codes.

quant-pharXiv:2603.18318•Mar 20, 2026•0 citations

Ordered Reliability Bits Guessing Random Additive Noise Decoding

Ken R. Duffy

Modern applications are driving demand for ultra-reliable low-latency communications, rekindling interest in the performance of short, high-rate error correcting codes. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can decode any moderate redundancy block-code. For a code of n bits, it avails of no more than ⎾log2(n)⏋ bits of code-book-independent quantized soft detection information per received bit to determine an accurate decoding while retaining the original algorithm’s suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide similar block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with low complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder.

arXiv:2001.00546•IEEE International Conference on Acoustics Speech and Signal Processing•May 13, 2021•78 citations

Turbo product decoding of cubic tensor codes

Sarah Khalifeh, Ken R. Duffy, Muriel Médard

Long, powerful soft detection forward error correction codes are typically constructed by concatenation of shorter component codes that are decoded through iterative Soft-Input Soft-Output (SISO) procedures. The current gold-standard is Low Density Parity Check (LDPC) codes, which are built from weak single parity check component codes that are capable of producing accurate SO. Due to the recent development of SISO decoders that produce highly accurate SO with codes that have multiple redundant bits, square product code constructions that can avail of more powerful component codes have been shown to be competitive with the LDPC codes in the 5G New Radio standard in terms of decoding performance while requiring fewer iterations to converge. Motivated by applications that require more powerful low-rate codes, in the present paper we explore the possibility of extending this design space by considering the construction and decoding of cubic tensor codes.

arXiv:2412.14017•Conference on Information Sciences and Systems•Mar 19, 2025•2 citations

We establish that a large, flexible class of long, high redundancy error correcting codes can be efficiently and accurately decoded with guessing random additive noise decoding (GRAND). Performance evaluation demonstrates that it is possible to construct simple product codes with lengths of approximately 200 to 4000 bits and rates between 0.2 and 0.8 that outperform low-density parity-check (LDPC) codes from the 5G New Radio standard in both AWGN and fading channels. The concatenated structure enables many desirable features, including: low-complexity hardware-friendly encoding and decoding; significant flexibility in length and rate through modularity; and high levels of parallelism in encoding and decoding that enable low latency. Central is the development of a method through which any soft-input (SI) GRAND algorithm can provide soft-output (SO) in the form of an accurate a-posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that each element of the list is correct. The distinguishing feature of soft-output GRAND (SOGRAND) is the provision of an estimate that the correct decoding has not been found, even when providing a single decoding. Per-block SO can be converted into accurate per-bit SO by a weighted sum that includes a term for the SI. Implementing SOGRAND adds negligible computation and memory to the existing decoding process, and using it results in a practical, low-latency alternative to LDPC codes.

Long forward error correction codes are typically constructed by concatenating shorter component codes that are then decoded through iterative Soft-Input Soft-Output (SISO) of their components. The recently introduced Soft Output Guessing Random Additive Noise Decoding (SOGRAND) has been shown to enable accurate SISO component decoding for a broad range of component codes. Here we establish that by specializing its SISO computation to Single Parity Check codes, SOGRAND offers an alternative existing Check Node (CN) update for decoding Low Density Parity Check codes. Simulation results demonstrate similar or better decoding performance than Gallager's sum-product algorithm and norm-min-sum, while offering two distinct low complexity, hardware friendly CN update algorithms.

Error correction techniques traditionally focus on the co-design of restricted code-structures in tandem with code-specific decoders that are computationally efficient when decoding long codes in hardware. Modern applications are, however, driving demand for ultra-reliable low-latency communications (URLLC), rekindling interest in the performance of shorter, higher-rate error correcting codes, and raising the possibility of revisiting universal, code-agnostic decoders. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can accurately decode any moderate redundancy block-code. It is designed with efficient circuit implementation in mind, and determines accurate decodings while retaining the original hard detection GRAND algorithm's suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide excellent soft decision block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with modest complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder. ORBGRAND offers the possibility of an accurate, energy efficient soft detection decoder suitable for delivering URLLC in a single hardware realization.

Maximum Likelihood (ML) decoding of forward error correction codes is known to be optimally accurate, but is not used in practice as it proves too challenging to efficiently implement. Here we propose a development of a previously described hard detection ML decoder called Guessing Random Additive Noise Decoding (GRAND). We introduce Soft GRAND (SGRAND), a ML decoder that fully avails of soft detection information and is suitable for use with any arbitrary high-rate, short-length block code. We assess SGRAND's performance on Cyclic Redundancy Check (CRC)-aided Polar (CA-Polar) codes, which will be used for all control channel communication in 5G New Radio (NR), comparing its accuracy with CRC-Aided Successive Cancellation List decoding (CA-SCL), a state-of-the-art soft-information decoder specific to CA-Polar codes.

We introduce a generalized low-density parity-check decoding framework for quantum Tanner codes utilizing soft-output guessing random additive noise decoding (SOGRAND). By soft-output decoding entire component codes rather than individual parity checks, we mitigate the effects of trapping sets and cycles, resulting in improved convergence. Because our decoder preserves the message passing structure of standard belief propagation (BP), it is compatible with many BP enhancements. We demonstrate this with OSD postprocessing, quaternary BP, and RelayBP, each yielding further gains. Standalone SOGRAND outperforms the standard BP+OSD baseline by over two orders of magnitude in logical error rate. In combination with the Relay principle, SOGRAND outperforms RelayBP by over one order of magnitude, providing a way forward for scalable decoding of the emerging class of Tanner-code-based quantum codes.

Modern applications are driving demand for ultra-reliable low-latency communications, rekindling interest in the performance of short, high-rate error correcting codes. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can decode any moderate redundancy block-code. For a code of n bits, it avails of no more than ⎾log2(n)⏋ bits of code-book-independent quantized soft detection information per received bit to determine an accurate decoding while retaining the original algorithm’s suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide similar block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with low complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder.

Long, powerful soft detection forward error correction codes are typically constructed by concatenation of shorter component codes that are decoded through iterative Soft-Input Soft-Output (SISO) procedures. The current gold-standard is Low Density Parity Check (LDPC) codes, which are built from weak single parity check component codes that are capable of producing accurate SO. Due to the recent development of SISO decoders that produce highly accurate SO with codes that have multiple redundant bits, square product code constructions that can avail of more powerful component codes have been shown to be competitive with the LDPC codes in the 5G New Radio standard in terms of decoding performance while requiring fewer iterations to converge. Motivated by applications that require more powerful low-rate codes, in the present paper we explore the possibility of extending this design space by considering the construction and decoding of cubic tensor codes.