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Exploiting the Degrees of Freedom: Multi-Dimensional Spatially-Coupled Codes Based on Gradient Descent

Ata Tanrıkulu, Mete Yıldırım, Ahmed Hareedy

Abstract

Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) 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 storage and 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, based on the gradient-descent (GD) algorithm, to design high performance MD codes where this challenge is addressed. In particular, we express the expected number of detrimental objects, 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 the finite-length (FL) algorithmic optimizer that produces the final MD-SC code. We adopt a recently-introduced Markov chain Monte Carlo (MCMC) FL algorithmic optimizer that is guided by the proposed GD algorithm. We apply our framework to various objects of interest. We start from simple short cycles, and then we develop the framework to address more sophisticated cycle concatenations, aiming at finer-grained optimization. We offer the theoretical analysis as well as the design algorithms. Next, we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower numbers of targeted detrimental objects compared with the available state-of-the-art. Moreover, we show that our GD-MD codes exhibit significant improvements in error-rate performance compared with MD-SC codes obtained by a uniform distribution.

Exploiting the Degrees of Freedom: Multi-Dimensional Spatially-Coupled Codes Based on Gradient Descent

Abstract

Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) 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 storage and 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, based on the gradient-descent (GD) algorithm, to design high performance MD codes where this challenge is addressed. In particular, we express the expected number of detrimental objects, 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 the finite-length (FL) algorithmic optimizer that produces the final MD-SC code. We adopt a recently-introduced Markov chain Monte Carlo (MCMC) FL algorithmic optimizer that is guided by the proposed GD algorithm. We apply our framework to various objects of interest. We start from simple short cycles, and then we develop the framework to address more sophisticated cycle concatenations, aiming at finer-grained optimization. We offer the theoretical analysis as well as the design algorithms. Next, we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower numbers of targeted detrimental objects compared with the available state-of-the-art. Moreover, we show that our GD-MD codes exhibit significant improvements in error-rate performance compared with MD-SC codes obtained by a uniform distribution.

Paper Structure

This paper contains 14 sections, 10 theorems, 60 equations, 10 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Novel Framework for Probabilistic MD-SC Code Design
  4. Probabilistic Derivations
  5. Gradient-Descent MD-SC Solution Form
  6. Outcomes Forecasted
  7. Theoretical Framework for Objects
  8. General Case Analysis
  9. Cycle Concatenation Analysis
  10. Design and Optimization Algorithms
  11. Multi-Dimensional Gradient-Descent Distributor
  12. Markov Chain Monte Carlo Finite-Length Optimizer
  13. Numerical Results and Comparisons
  14. Conclusion

Key Result

Lemma 1

Each cycle-$2g$ in the Tanner graph of an MD-SC code corresponds to a cycle-$2g$ candidate in the protograph base matrix, $\mathbf{H}^{\textup{g}}$.A cycle candidate is a way of traversing a protograph pattern to reach a cycle after partitioning/relocations/lifting (see channel_aware). We label a cy This cycle candidate becomes a cycle-$2g$ candidate in the MD protograph, i.e., remains active, if

Figures (10)

  • Figure 1: Seven (out of nine) protograph patterns that can result in cycles-$8$ in the Tanner graph after partitioning/relocations/lifting. The transposes of $P_2$ and $P_4$ are excluded from the list for brevity. Light green squares are the non-zero entries specifying the cycle-candidate edges.
  • Figure 2: Bipartite graphs of $6$-$6$ (top left), $6$-$8$ (top right), and $8$-$8$ (bottom) configurations.
  • Figure 3: Bipartite graph of a $6$-$8$ configuration (left) and matrix representation of one of its object patterns (right).
  • Figure 4: Example object pattern class matrix representations of the $6$-$6$, $6$-$8$, and $8$-$8$ configurations grouped by the values in $( |\mathcal{E}|,|\mathcal{V}|, |\mathcal{C}|)$.
  • Figure 7: FER curves of GD-GD/UNF-UNF MD Code 8 over the AWGNC. Both curves correspond to codes with parameters $\left(\gamma, \kappa, z, L, m, M\right) = \left(4,17,7,15,4,5\right)$.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 29 more