Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Balanced Tree Transformation to Reduce GRAND Queries

Lukas Rapp, Jiewei Feng, Muriel Médard, Ken R. Duffy

TL;DR

This work tackles reducing GRAND-based decoding queries by introducing Balanced Tree Transformation (BTT), which rewrites a code's parity-check matrix into a Tree Structure via a random invertible transform. This enables extraction of multiple disjoint noise-pattern constraints, enhancing Segmented GRAND's ability to skip invalid patterns while preserving code equivalence. Theoretical guarantees (thm_uniform_matrix) and simulations on BCH(127,106) demonstrate up to ~2x query reduction per added constraint with only minor SNR losses at target BLER, highlighting substantial practical speedups for near-ML decoding of arbitrary binary codes. The approach promises broader applicability to accelerate GRAND decoders without structural changes to the underlying codes.

Abstract

Guessing Random Additive Noise Decoding (GRAND) and its variants, known for their near-maximum likelihood performance, have been introduced in recent years. One such variant, Segmented GRAND, reduces decoding complexity by generating only noise patterns that meet specific constraints imposed by the linear code. In this paper, we introduce a new method to efficiently derive multiple constraints from the parity check matrix. By applying a random invertible linear transformation and reorganizing the matrix into a tree structure, we extract up to log2(n) constraints, reducing the number of decoding queries while maintaining the structure of the original code for a code length of n. We validate the method through theoretical analysis and experimental simulations.

A Balanced Tree Transformation to Reduce GRAND Queries

TL;DR

This work tackles reducing GRAND-based decoding queries by introducing Balanced Tree Transformation (BTT), which rewrites a code's parity-check matrix into a Tree Structure via a random invertible transform. This enables extraction of multiple disjoint noise-pattern constraints, enhancing Segmented GRAND's ability to skip invalid patterns while preserving code equivalence. Theoretical guarantees (thm_uniform_matrix) and simulations on BCH(127,106) demonstrate up to ~2x query reduction per added constraint with only minor SNR losses at target BLER, highlighting substantial practical speedups for near-ML decoding of arbitrary binary codes. The approach promises broader applicability to accelerate GRAND decoders without structural changes to the underlying codes.

Abstract

Guessing Random Additive Noise Decoding (GRAND) and its variants, known for their near-maximum likelihood performance, have been introduced in recent years. One such variant, Segmented GRAND, reduces decoding complexity by generating only noise patterns that meet specific constraints imposed by the linear code. In this paper, we introduce a new method to efficiently derive multiple constraints from the parity check matrix. By applying a random invertible linear transformation and reorganizing the matrix into a tree structure, we extract up to log2(n) constraints, reducing the number of decoding queries while maintaining the structure of the original code for a code length of n. We validate the method through theoretical analysis and experimental simulations.

Paper Structure

This paper contains 7 sections, 1 theorem, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tree Structure and balanced tree transformation
  4. Application of Tree Structure
  5. Simulation Results
  6. Conclusion
  7. Acknowledgment

Key Result

Theorem 1

Let $H^{m,\kappa m}$ be a given $m\times \kappa m$ binary matrix, for $\kappa\in \mathbb{N}$. Separate $H^{m,\kappa m}$ into $\kappa$ non-overlapping sub-matrices with dimension $m \times m$ and denote each of the sub-matrices as $H^{m,\kappa m}_{(j)}$. Let $\mathcal{H}^{m,\kappa m}:= \{H^{m,\kappa

Figures (3)

  • Figure 1: Example of Tree Structure. For each row, adjacent entries with $1$ are highlighted in orange, while adjacent entries with $0$ are highlighted in blue. Each rectangle represents a subset of the columns, for example, the orange rectangle in the first row represents $\mathcal{L}_1$ and the second rectangle in the second row represents $\mathcal{L}_{0,1}$. As demonstrated in the figure, the columns are divided into two disjoint sets based on the first row, and then each of the subsets is further divided into subsets by the second row and the third row as well. This exhibits a branching Tree Structure and hence we call this the Tree Structure. Note that as demonstrated in the figure, the number of subsets that the bits can be divided into is bounded above by $2^m$ and $\log_2n$, where $m$ is the number of rows of $H$ and $n$ is the number of columns of $H$.
  • Figure 2: Example of rewriting BCH(127, 106) code.
  • Figure 3: Example for query number reduction using the Balanced Tree Transformation on BCH (127,106) code.

Theorems & Definitions (2)

  • Theorem 1
  • proof