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Adapt or Regress: Rate-Memory-Compatible Spatially-Coupled Codes

Bade Aksoy, Doğukan Özbayrak, Ahmed Hareedy

TL;DR

The paper addresses the challenge of designing rate-compatible LDPC SC codes by introducing rate-memory-compatible SC (RMC-SC) codes that increase code memory to achieve rate adaptation. It develops a probabilistic, multi-stage design framework in which a fixed protograph is augmented by new components and coupled via a gradient-descent optimization of the new-edge distribution $oldsymbol{q}$, yielding an overall distribution $oldsymbol{u}$ that minimizes short-cycle counts. To realize practical finite-length performance, the authors deploy a Markov chain Monte Carlo (MC2) optimizer for both partitioning and lifting across design stages, achieving substantially lower cycle counts and strong FER gains (e.g., orders of magnitude improvements over straightforward RC-SC designs) with notable hardware reductions. The results demonstrate that RMC-SC codes can adapt to varying channel conditions and device aging while delivering significant reliability and hardware-efficiency benefits, and the framework can extend to other rate-compatibility schemes, with future work including ML integration and threshold analysis.

Abstract

Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) codes that have excellent performance thanks to the degrees of freedom they offer. An SC code is designed by partitioning a base matrix into components, the number of which implies the code memory, then coupling and lifting them. In the same system, various error-correction coding schemes are typically needed. For example, in wireless communication standards, several channel conditions and data rates should be supported. In storage and computing systems, stronger codes should be adopted as the device ages. Adaptive code design enables switching from one code to another when needed, ensuring reliability while reducing hardware cost. In this paper, we introduce a class of reconfigurable SC codes named rate-memory-compatible SC (RMC-SC) codes, which we design probabilistically. In particular, rate compatibility in RMC-SC codes is achieved via increasing the SC code memory, which also makes the codes memory-compatible and improves performance. We express the expected number of short cycles in the SC code protograph as a function of the fixed probability distribution characterizing the already-designed SC code as well as the unknown distribution characterizing the additional components. We use the gradient-descent algorithm to find a locally-optimal distribution, in terms of cycle count, for the new components. The method can be recursively used to design any number of SC codes needed, and we show how to extend it to other cases. Next, we perform the finite-length optimization using a Markov chain Monte Carlo (MC$^2$) approach that we update to design the proposed RMC-SC codes. Experimental results demonstrate significant reductions in cycle counts and remarkable performance gains achieved by RMC-SC codes compared with a literature-based straightforward scheme.

Adapt or Regress: Rate-Memory-Compatible Spatially-Coupled Codes

TL;DR

The paper addresses the challenge of designing rate-compatible LDPC SC codes by introducing rate-memory-compatible SC (RMC-SC) codes that increase code memory to achieve rate adaptation. It develops a probabilistic, multi-stage design framework in which a fixed protograph is augmented by new components and coupled via a gradient-descent optimization of the new-edge distribution , yielding an overall distribution that minimizes short-cycle counts. To realize practical finite-length performance, the authors deploy a Markov chain Monte Carlo (MC2) optimizer for both partitioning and lifting across design stages, achieving substantially lower cycle counts and strong FER gains (e.g., orders of magnitude improvements over straightforward RC-SC designs) with notable hardware reductions. The results demonstrate that RMC-SC codes can adapt to varying channel conditions and device aging while delivering significant reliability and hardware-efficiency benefits, and the framework can extend to other rate-compatibility schemes, with future work including ML integration and threshold analysis.

Abstract

Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) codes that have excellent performance thanks to the degrees of freedom they offer. An SC code is designed by partitioning a base matrix into components, the number of which implies the code memory, then coupling and lifting them. In the same system, various error-correction coding schemes are typically needed. For example, in wireless communication standards, several channel conditions and data rates should be supported. In storage and computing systems, stronger codes should be adopted as the device ages. Adaptive code design enables switching from one code to another when needed, ensuring reliability while reducing hardware cost. In this paper, we introduce a class of reconfigurable SC codes named rate-memory-compatible SC (RMC-SC) codes, which we design probabilistically. In particular, rate compatibility in RMC-SC codes is achieved via increasing the SC code memory, which also makes the codes memory-compatible and improves performance. We express the expected number of short cycles in the SC code protograph as a function of the fixed probability distribution characterizing the already-designed SC code as well as the unknown distribution characterizing the additional components. We use the gradient-descent algorithm to find a locally-optimal distribution, in terms of cycle count, for the new components. The method can be recursively used to design any number of SC codes needed, and we show how to extend it to other cases. Next, we perform the finite-length optimization using a Markov chain Monte Carlo (MC) approach that we update to design the proposed RMC-SC codes. Experimental results demonstrate significant reductions in cycle counts and remarkable performance gains achieved by RMC-SC codes compared with a literature-based straightforward scheme.

Paper Structure

This paper contains 7 sections, 5 theorems, 26 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. RMC-SC Code Design Idea
  3. Probabilistic Analysis and Extension
  4. Gradient-Descent Distribution
  5. Finite-Length MCMC Optimization
  6. Experimental Results
  7. Conclusion

Key Result

Theorem 1

Consider an RMC-SC code that has $\mathbf{H}_{\textup{f}}$, with $r_{\textup{f}}$ and $m_{\textup{f}}$, as well as $\mathbf{H}_{\textup{n}}$, with $r_{\textup{n}}$ and $m_{\textup{n}}$. Denote the expected number of cycle-$2\ell$ candidates that has $2\ell$ distinct entries in the RMC-SC protograph where $A_{2\ell}$ is the number of all such cycle-$2\ell$ candidates in the all-one matrix $\mathbf

Figures (4)

  • Figure 1: The only cycle-$6$ protograph pattern and its candidate (top left) as well as three (out of five) cycle-$8$ protograph patterns and one candidate for each. Light red squares are the non-zero entries of the cycle-candidate.
  • Figure 2: Probability distributions of RMC-SC codes at different design stages as well as the optimal distribution for $m=15$.
  • Figure 3: FER versus SNR curves of RMC-SC and SF-SC codes designed with the first group of parameters, $(\gamma, \kappa, z, L, m_{\textup{n},0}, m_{\textup{n},1}, m_{\textup{n},2})=(7, 23, 23, 12, 6, 2, 3)$, over the AWGNC.
  • Figure 4: FER versus crossover probability curves of RMC-SC and SF-SC codes designed with the second group of parameters, $(\gamma, \kappa, z, L, m_{\textup{n},0}, m_{\textup{n},1}, m_{\textup{n},2})=(7, 35, 29, 16, 8, 3, 4)$, over the BSC.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 3 more