[Paper Review] Band Splitting Permutations for Spatially Coupled LDPC Codes Enhancing Burst Erasure Immunity

Hiroki MORI, Tadashi Wadayama|arXiv (Cornell University)|Jan 19, 2015

Cooperative Communication and Network Coding1 references3 citations

TL;DR

This paper proposes band splitting permutations (BSP) for spatially coupled LDPC codes to enhance burst erasure correction by restructuring the base matrix into multiple diagonal bands, significantly increasing the span of stopping sets. The method achieves asymptotic optimality in burst erasure correction, with the maximal correctable burst ratio approaching $\lambda_{\text{max}} \simeq 1/k$ where $k = r/l$, outperforming conventional SC codes in burst erasure channels.

ABSTRACT

It is well known that spatially coupled (SC) codes with erasure-BP decoding have powerful error correcting capability over memoryless erasure channels. However, the decoding performance of SC-codes significantly degrades when they are used over burst erasure channels. In this paper, we propose band splitting permutations (BSP) suitable for $(l,r,L)$ SC-codes. The BSP splits a diagonal band in a base matrix into multiple bands in order to enhance the span of the stopping sets in the base matrix. As theoretical performance guarantees, lower and upper bounds on the maximal burst correctable length of the permuted $(l,r,L)$ SC-codes are presented. Those bounds indicate that the maximal correctable burst ratio of the permuted SC-codes converges to 1/k where k=r/l. This implies the asymptotic optimality of the permuted SC-codes in terms of burst erasure correction.

Motivation & Objective

• To address the degradation in decoding performance of spatially coupled LDPC codes over burst erasure channels, where conventional SC codes suffer due to disrupted reliability wave propagation.
• To improve the burst erasure correcting capability of $(l,r,L)$ SC-codes by reordering the parity check matrix through column permutations that increase stopping set span.
• To theoretically bound the maximal correctable burst length of permuted SC-codes and demonstrate their asymptotic optimality in burst erasure correction.
• To design a systematic permutation method—band splitting permutations (BSP)—that enhances burst immunity without degrading performance on memoryless erasure channels.

Proposed method

• Band splitting permutations (BSP) are applied to the base matrix of $(l,r,L)$ SC-codes, transforming a single diagonal band into multiple distributed diagonal bands to increase the span of stopping sets.
• The method involves reordering columns of the base matrix such that irreducible stopping sets span at least $L+1$ positions, ensuring longer minimum burst length correction.
• Theoretical analysis derives lower and upper bounds on the maximal correctable burst length, showing $\frac{L-1}{kL} < \lambda_{\text{max}} < \frac{L+1}{kL}$, which converge to $1/k$ as $L \to \infty$.
• The construction preserves the original code rate and design rate $R(l,r,L) = 1 - \frac{1}{k} - \frac{l-1}{kL}$, ensuring no performance loss on memoryless channels.
• The permutation is designed to maintain a one-to-one correspondence between stopping sets in the original and permuted base matrices, enabling theoretical analysis of stopping set structure.
• Computer experiments validate the theoretical bounds, comparing BSP-permuted codes with randomly permuted and conventional SC codes, showing superior burst correction.

Experimental results

Research questions

• RQ1 Can a systematic column permutation of the base matrix of $(l,r,L)$ SC-codes improve their burst erasure correction capability without degrading memoryless erasure performance?
• RQ2 What is the theoretical upper and lower bound on the maximal correctable burst length of permuted SC-codes using band splitting permutations?
• RQ3 Does the maximal correctable burst ratio $\lambda_{\text{max}}$ of BSP-permuted SC-codes approach $1/k$ asymptotically as $L \to \infty$, indicating optimality?
• RQ4 How does the burst erasure correction performance of BSP-permuted SC-codes compare to conventional and randomly permuted SC-codes in finite-length settings?

Key findings

• The maximal correctable burst length of BSP-permuted $(l,r,L)$ SC-codes is bounded by $\frac{L-1}{kL} < \lambda_{\text{max}} < \frac{L+1}{kL}$, with $\lambda_{\text{max}} \to 1/k$ as $L \to \infty$.
• For $(3,6,L)$ codes, $\lambda_{\text{max}}$ of BSP-permuted codes converges to 0.5, while the BP threshold is 0.488 at $L=128$, indicating superior burst correction.
• The permuted SC-codes achieve asymptotic optimality in burst erasure correction, satisfying $\lim_{L\to\infty}(\lambda_{\text{max}} + R(l,r,L)) = 1$, the theoretical limit for any binary linear code.
• Computer experiments show that $\lambda_{\text{max}}$ of randomly permuted SC-codes is significantly lower than that of BSP-permuted codes, proving the non-triviality of the proposed design.
• The span of irreducible stopping sets in the permuted base matrix is at least $L+1$, which directly enables longer correctable burst lengths.
• The proposed BSP method enhances burst erasure immunity without altering the code rate or degrading performance on memoryless erasure channels.