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Leveraging Code Structure to Improve Soft Output for GRAND, GCD, OSD, and SCL

Jiewei Feng, Ken R. Duffy, Muriel Médard

TL;DR

This work addresses the problem of obtaining accurate blockwise soft output (SO) from decoders that may not rely on large lists. By incorporating binary linear codebook constraints, particularly for even codes, the authors refine the posterior likelihood estimates used to produce SO across GRAND, GCD, OSD, and SCL decoding, and they quantify SO quality with a Brier Score. Theoretical analysis shows that the gains from linear-code constraints are present but modest, while empirical results on eBCH codes demonstrate that SO-GRAND and SO-GCD with small list sizes can closely match ML SO. The findings suggest practical benefits for early stopping and iterative decoding with reduced complexity, enabled by more calibrated and refined SO estimates.

Abstract

In addition to a proposed codeword, error correction decoders that provide blockwise soft output (SO) return an estimate of the likelihood that the decoding is correct. Following Forney, such estimates are traditionally only possible for list decoders where the soft output is the likelihood that a decoding is correct given it is assumed to be in the list. Recently, it has been established that Guessing Random Additive Noise Decoding (GRAND), Guessing Codeword Decoding (GCD), Ordered Statistics Decoding (OSD), and Successive Cancellation List (SCL) decoding can provide more accurate soft output, even without list decoding. Central to the improvement is a per-decoding estimate of the likelihood that a decoding has not been found that can be readily calculated during the decoding process. Here we explore how linear codebook constraints can be employed to further enhance the precision of such SO. We evaluate performance by adapting a forecasting statistic called the Brier Score. Results indicate that the SO generated by the approach is essentially as accurate as the maximum a posteriori estimate.

Leveraging Code Structure to Improve Soft Output for GRAND, GCD, OSD, and SCL

TL;DR

This work addresses the problem of obtaining accurate blockwise soft output (SO) from decoders that may not rely on large lists. By incorporating binary linear codebook constraints, particularly for even codes, the authors refine the posterior likelihood estimates used to produce SO across GRAND, GCD, OSD, and SCL decoding, and they quantify SO quality with a Brier Score. Theoretical analysis shows that the gains from linear-code constraints are present but modest, while empirical results on eBCH codes demonstrate that SO-GRAND and SO-GCD with small list sizes can closely match ML SO. The findings suggest practical benefits for early stopping and iterative decoding with reduced complexity, enabled by more calibrated and refined SO estimates.

Abstract

In addition to a proposed codeword, error correction decoders that provide blockwise soft output (SO) return an estimate of the likelihood that the decoding is correct. Following Forney, such estimates are traditionally only possible for list decoders where the soft output is the likelihood that a decoding is correct given it is assumed to be in the list. Recently, it has been established that Guessing Random Additive Noise Decoding (GRAND), Guessing Codeword Decoding (GCD), Ordered Statistics Decoding (OSD), and Successive Cancellation List (SCL) decoding can provide more accurate soft output, even without list decoding. Central to the improvement is a per-decoding estimate of the likelihood that a decoding has not been found that can be readily calculated during the decoding process. Here we explore how linear codebook constraints can be employed to further enhance the precision of such SO. We evaluate performance by adapting a forecasting statistic called the Brier Score. Results indicate that the SO generated by the approach is essentially as accurate as the maximum a posteriori estimate.

Paper Structure

This paper contains 6 sections, 3 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Soft output formulae
  3. Leveraging Binary Linear Codebook Structure for SO
  4. SO Quality - Theory
  5. SO Quality Quantification
  6. Conclusions

Key Result

Theorem 1

Assuming a GRAND decoder is used to decode an even code. Define $I_{i}=1$ if the $i$-th queried noise effect has the same parity as $y^n$ and $I_{i}=0$ if otherwise. Since the code is even, assuming that the unvisited codewords are uniformly distributed within the noise effects that have not been ge where $\phi(j)=p_{Z^n|R^n}(z^{n,j}|r^n)$ and

Figures (2)

  • Figure 1: Example of BS vs Eb/N0 using eBCH(16, 11) code.
  • Figure 2: Example of BS using eBCH(16, 11) code.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof