Spatially-coupled Split-component Codes with Iterative Algebraic Decoding

TL;DR

This paper introduces a class of spatially-coupled split-component error-correcting codes, analyzes their iterative algebraic decoding performance over different channels, and extends the analysis to mixture ensembles and beyond bounded-distance decoding.

Contribution

It generalizes existing codes like staircase and braided codes, providing a comprehensive analysis framework for their decoding thresholds and performance.

Findings

Derived a vector recursion for residual graph evolution.

Identified decoding thresholds using potential function analysis.

Extended analysis to mixture ensembles and beyond bounded-distance decoding.

Abstract

We analyze a class of high performance, low decoding-data-flow error-correcting codes suitable for high bit-rate optical-fiber communication systems. A spatially-coupled split-component ensemble is defined, generalizing from the most important codes of this class, staircase codes and braided block codes, and preserving a deterministic partitioning of component-code bits over code blocks. Our analysis focuses on low-complexity iterative algebraic decoding, which, for the binary erasure channel, is equivalent to a generalization of the peeling decoder. Using the differential equation method, we derive a vector recursion that tracks the expected residual graph evolution throughout the decoding process. The threshold of the recursion is found using potential function analysis. We generalize the analysis to mixture ensembles consisting of more than one type of component code, which provide increased flexibility of ensemble parameters and can improve performance. The analysis extends to the binary symmetric channel by assuming mis-correction-free component-code decoding. Simple upper-bounds on the number of errors correctable by the ensemble are derived. Finally, we analyze the threshold of spatially-coupled split-component ensembles under beyond bounded-distance component decoding.