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Soft-output (SO) GRAND and Iterative Decoding to Outperform LDPCs

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

TL;DR

The paper introduces SOGRAND, a soft-output extension to the GRAND decoding framework, enabling accurate per-block and per-bit a-posteriori probabilities for long, high-redundancy codes. By deriving exact and approximated SO expressions and leveraging GRAND’s codebook-agnostic decoding, SOGRAND demonstrates that simple product and GLDPC codes can outperform 5G NR LDPC codes in both AWGN and block-fading channels, with lower latency and highly parallelizable hardware implementation. The approach provides a practical, flexible alternative to LDPC for long codes, offering robust soft information and scalable encoding schemes. Overall, SOGRAND broadens the design space for efficient, low-latency decoders of powerful, modular codes suitable for next-generation wireless systems.

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. That 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.

Soft-output (SO) GRAND and Iterative Decoding to Outperform LDPCs

TL;DR

The paper introduces SOGRAND, a soft-output extension to the GRAND decoding framework, enabling accurate per-block and per-bit a-posteriori probabilities for long, high-redundancy codes. By deriving exact and approximated SO expressions and leveraging GRAND’s codebook-agnostic decoding, SOGRAND demonstrates that simple product and GLDPC codes can outperform 5G NR LDPC codes in both AWGN and block-fading channels, with lower latency and highly parallelizable hardware implementation. The approach provides a practical, flexible alternative to LDPC for long codes, offering robust soft information and scalable encoding schemes. Overall, SOGRAND broadens the design space for efficient, low-latency decoders of powerful, modular codes suitable for next-generation wireless systems.

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. That 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.
Paper Structure (15 sections, 2 theorems, 31 equations, 13 figures)

This paper contains 15 sections, 2 theorems, 31 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Per-block SO
  3. Preliminaries
  4. Notations
  5. Background on GRAND
  6. Background on SO
  7. GRAND Per Block SO
  8. SO accuracy
  9. Blockwise SO accuracy
  10. Bitwise SO accuracy
  11. Long Code Performance Evaluation
  12. AWGN Channels
  13. Block Fading Channels
  14. Encoding Complexity.
  15. Discussion

Key Result

Theorem 1

Given the soft information $R^n=r^n$ that determines the query order, let $G(N^n)$ be the number of codebook queries until the noise effect sequence $N^n$ is identified. Let $W_1,\ldots,W_{2^k-1}$ be selected uniformly at random without replacement from $\{1,\ldots,2^n-1\}$ and define their rank-ord which is the vector ${q_{1}^{L}}$ but with the entry $q_i$ omitted and one subtracted for all entri

Figures (13)

  • Figure 1: AWGN BLER performance of the $(1024, 441)$ 5G LDPC with maximum iteration number $50$ as compared to a $(1024, 441)=(32,21)^2$ dRM product code decoded two ways. First, turbo-decoded pyndiah_1998, with $\alpha$ and $\beta$ parameters taken from there and maximum iteration number $20$, but using $1$line-ORBGRAND for list decoding with list size $L=4$ as in galligan2023_block. Second, turbo-decoded using $\alpha=0.5$ and SOGRAND with max iteration $20$, where lists are added to until $L=4$ or the predicted list-BLER is below $10^{-5}$.
  • Figure 2: Predicted list-BLER based on soft-output vs. empirical list-BLER ($L^\prime=4$): SOGRAND with $L=4$, Forney with $L=5$, $E_b/N_0=2$.
  • Figure 3: Soft-Input Soft-Output (SISO) decoder schematic.
  • Figure 4: SO predicted vs. empirical BER: $L=4$, $E_b/N_0=2$.
  • Figure 5: AWGN performance of $(256, 121)$ 5G LDPC with max. iterations $50$ and the $(256, 121)$ 5G CA-Polar ($24$-bits CRC) decoded with CA-SCL ($L=16$) as compared to a $(256, 121)=(16,11)^2$ eBCH product code decoded with SOGRAND, with $\alpha=0.5$ and maximum iteration number $20$, where lists are added to until $L=4$ or the predicted list-BLER is below $10^{-5}$. Upper panel: BLER and BER. Middle panel: average number of queries per-bit until a decoding, where parallelized assumes all rows/columns are decoded in parallel. Lower panel: average number of iterations.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 1: GRAND list decoding APPs for a uniformly random codebook
  • proof
  • Corollary 1: Approximate APPs for a random codebook
  • proof