Coding for Parallel Channels: Gallager Bounds and Applications to Repeat-Accumulate Codes

TL;DR

This paper extends Gallager bounds to parallel channels, providing new upper bounds on ML decoding error probabilities and applying them to analyze turbo-like and repeat-accumulate codes.

Contribution

It generalizes the DS2 bounds to parallel channels and connects them with Gallager bounds, offering improved error probability bounds for turbo-like code ensembles.

Findings

Derived new upper bounds for ML decoding error probability.

Connected DS2 bounds with Gallager bounds for parallel channels.

Applied bounds to analyze repeat-accumulate code ensembles.

Abstract

This paper is focused on the performance analysis of binary linear block codes (or ensembles) whose transmission takes place over independent and memoryless parallel channels. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived. The framework of the second version of the Duman and Salehi (DS2) bounds is generalized to the case of parallel channels, along with the derivation of optimized tilting measures. The connection between the generalized DS2 and the 1961 Gallager bounds, known previously for a single channel, is revisited for the case of parallel channels. The new bounds are used to obtain improved inner bounds on the attainable channel regions under ML decoding. These improved bounds are applied to ensembles of turbo-like codes, focusing on repeat-accumulate codes and their recent variations.