Density Evolution for Deterministic Generalized Product Codes on the Binary Erasure Channel at High Rates

TL;DR

This paper analyzes the asymptotic performance of deterministic generalized product codes over the binary erasure channel using density evolution, providing insights into their decoding behavior and design at high rates.

Contribution

It introduces a deterministic construction of GPCs, derives density evolution equations, and applies these results to optimize irregular GPC designs.

Findings

Density evolution equations characterize asymptotic decoding performance.

Equivalent to peeling algorithms on sparse inhomogeneous graphs.

Design guidelines for irregular GPCs with mixed component codes.

Abstract

Generalized product codes (GPCs) are extensions of product codes (PCs) where code symbols are protected by two component codes but not necessarily arranged in a rectangular array. We consider a deterministic construction of GPCs (as opposed to randomized code ensembles) and analyze the asymptotic performance over the binary erasure channel under iterative decoding. Our code construction encompasses several classes of GPCs previously proposed in the literature, such as irregular PCs, block-wise braided codes, and staircase codes. It is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity. We show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph. Using a convergence result for these graphs, we derive the density evolution equations that characterize the asymptotic decoding performance. As an application, we discuss the design of irregular GPCs employing a mixture of component codes with different erasure-correcting capabilities.