Deterministic and Ensemble-Based Spatially-Coupled Product Codes

TL;DR

This paper analyzes a family of deterministic and ensemble-based spatially-coupled product codes, deriving their performance over erasure channels and showing their asymptotic behavior matches that of related ensembles.

Contribution

It introduces a parametrized family of generalized product codes, including staircase and braided codes, and derives their density evolution equations for iterative decoding analysis.

Findings

Density evolution equations are rigorously derived for the deterministic codes.

Deterministic braided codes share the same DE as related ensembles.

Performance over binary erasure channel is characterized asymptotically.

Abstract

Several authors have proposed spatially-coupled (or convolutional-like) variants of product codes (PCs). In this paper, we focus on a parametrized family of generalized PCs that recovers some of these codes (e.g., staircase and block-wise braided codes) as special cases and study the iterative decoding performance over the binary erasure channel. Even though our code construction is deterministic (and not based on a randomized ensemble), we show that it is still possible to rigorously derive the density evolution (DE) equations that govern the asymptotic performance. The obtained DE equations are then compared to those for a related spatially-coupled PC ensemble. In particular, we show that there exists a family of (deterministic) braided codes that follows the same DE equation as the ensemble, for any spatial length and coupling width.