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Probabilistic Design of Multi-Dimensional Spatially-Coupled Codes

Canberk İrimağzı, Ata Tanrıkulu, Ahmed Hareedy

TL;DR

The paper tackles reducing short cycles in multi-dimensional spatially-coupled (MD-SC) LDPC codes by introducing a probabilistic framework that uses a joint probability-distribution matrix $\mathbf{P}$ to govern MD relocations. It derives the cycle-6 and cycle-8 metrics $P_6(\mathbf{p}^{\textup{con}})$ and $N_8(\mathbf{p}^{\textup{con}})$, and furnishes a gradient-descent solution form (MD-GRADE) together with a finite-length optimizer (FL-AO) to realize high-performance MD-SC codes. The approach yields locally-optimal relocation patterns that dramatically reduce short-cycle counts (e.g., $N_6$, $N_8$) and converge in few iterations, enabling scalable design for high-memory MD-SC codes. Experiments across multiple base SC/TC codes show substantial cycle reductions (often exceeding 90% in some cases) and competitive finite-length performance, illustrating the practical impact of probabilistic MD-SC design for data storage and transmission systems.

Abstract

Because of their excellent asymptotic and finite-length performance, spatially-coupled (SC) codes are a class of low-density parity-check codes that is gaining increasing attention. Multi-dimensional (MD) SC codes are constructed by connecting copies of an SC code via relocations in order to mitigate various sources of non-uniformity and improve performance in many data storage and data transmission systems. As the number of degrees of freedom in the MD-SC code design increases, appropriately exploiting them becomes more difficult because of the complexity growth of the design process. In this paper, we propose a probabilistic framework for the MD-SC code design, which is based on the gradient-descent (GD) algorithm, to design better MD codes and address this challenge. In particular, we express the expected number of short cycles, which we seek to minimize, in the graph representation of the code in terms of entries of a probability-distribution matrix that characterizes the MD-SC code design. We then find a locally-optimal probability distribution, which serves as the starting point of a finite-length algorithmic optimizer that produces the final MD-SC code. We offer the theoretical analysis as well as the algorithms, and we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower short cycle numbers compared with the available state-of-the-art. Moreover, our algorithms converge on solutions in few iterations, which confirms the complexity reduction as a result of limiting the search space via the locally-optimal GD-MD distributions.

Probabilistic Design of Multi-Dimensional Spatially-Coupled Codes

TL;DR

The paper tackles reducing short cycles in multi-dimensional spatially-coupled (MD-SC) LDPC codes by introducing a probabilistic framework that uses a joint probability-distribution matrix to govern MD relocations. It derives the cycle-6 and cycle-8 metrics and , and furnishes a gradient-descent solution form (MD-GRADE) together with a finite-length optimizer (FL-AO) to realize high-performance MD-SC codes. The approach yields locally-optimal relocation patterns that dramatically reduce short-cycle counts (e.g., , ) and converge in few iterations, enabling scalable design for high-memory MD-SC codes. Experiments across multiple base SC/TC codes show substantial cycle reductions (often exceeding 90% in some cases) and competitive finite-length performance, illustrating the practical impact of probabilistic MD-SC design for data storage and transmission systems.

Abstract

Because of their excellent asymptotic and finite-length performance, spatially-coupled (SC) codes are a class of low-density parity-check codes that is gaining increasing attention. Multi-dimensional (MD) SC codes are constructed by connecting copies of an SC code via relocations in order to mitigate various sources of non-uniformity and improve performance in many data storage and data transmission systems. As the number of degrees of freedom in the MD-SC code design increases, appropriately exploiting them becomes more difficult because of the complexity growth of the design process. In this paper, we propose a probabilistic framework for the MD-SC code design, which is based on the gradient-descent (GD) algorithm, to design better MD codes and address this challenge. In particular, we express the expected number of short cycles, which we seek to minimize, in the graph representation of the code in terms of entries of a probability-distribution matrix that characterizes the MD-SC code design. We then find a locally-optimal probability distribution, which serves as the starting point of a finite-length algorithmic optimizer that produces the final MD-SC code. We offer the theoretical analysis as well as the algorithms, and we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower short cycle numbers compared with the available state-of-the-art. Moreover, our algorithms converge on solutions in few iterations, which confirms the complexity reduction as a result of limiting the search space via the locally-optimal GD-MD distributions.
Paper Structure (13 sections, 5 theorems, 31 equations, 5 figures, 2 tables, 3 algorithms)

This paper contains 13 sections, 5 theorems, 31 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Novel Framework for Probabilistic MD-SC Code Design
  4. Probabilistic Metrics
  5. Gradient-Descent MD-SC Solution Form
  6. Outcomes Forecasted
  7. Design and Optimization Algorithms
  8. Multi-Dimensional Gradient-Descent Distributor
  9. Probabilistic Design of GD-MD Codes
  10. Experimental Results and Comparisons
  11. Conclusion
  12. Appendix
  13. Acknowledgements

Key Result

Theorem 1

Let $[\cdot]_{i,j}$ denote the coefficient of $X^iY^j$ in a two-variable polynomial. Denote by $P_6(\mathbf{p}^{\textup{con}})$ the probability of a cycle-$6$ candidate in the base matrix becoming a cycle-$6$ candidate in the MD protograph $\mathbf{H}_{\textup{MD}}^{\textup{g}}$ under random partiti Thus, the expected number $N_6(\mathbf{p}^{\textup{con}})$ of cycle-$6$ candidates in $\mathbf{H}_{

Figures (5)

  • Figure 1: Seven (out of nine) protograph patterns that can result in cycles-$8$ in the Tanner graph after lifting. The transposes of $P_2$ and $P_4$ are excluded from the list. Light green squares are the non-zero entries specifying the cycle-candidate edges.
  • Figure 2: Partitioning matrix (top) and lifting matrix (bottom) of SC Code 1 with parameters $(\gamma,\kappa,z,L,m) = (4,17,17,10,1)$.
  • Figure 3: Initial probability-distribution matrix (top) and relocation matrix (bottom) of GD-MD Code 1.1 with constituent SC Code 1.
  • Figure 4: Partitioning matrix (top) and lifting matrix (bottom) of SC Code 3 with $(\gamma,\kappa,z,L,m)=(3,19,23,10,2)$.
  • Figure 5: Initial probability-distribution matrix (top) and relocation matrix (bottom) of GD-MD Code 3.1 with constituent SC Code 3.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • ...and 10 more