Density Evolution for Deterministic Generalized Product Codes with Higher-Order Modulation

TL;DR

This paper extends density evolution analysis for deterministic generalized product codes to parallel erasure channels with different probabilities, aiding in the design of coded modulation schemes with higher-order constellations.

Contribution

It introduces a density evolution framework for GPCs over parallel BECs and applies it to optimize bit mapping in higher-order modulation systems.

Findings

Extended density evolution for GPCs over parallel BECs.

Predicted code performance in coded modulation with higher-order constellations.

Guidelines for designing bit mappers in such systems.

Abstract

Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case where transmission takes place over parallel BECs with different erasure probabilities. We use this model to predict the code performance in a coded modulation setup with higher-order signal constellations. We also discuss the design of the bit mapper that determines the allocation of the coded bits to the modulation bits of the signal constellation.