Why We Can Not Surpass Capacity: The Matching Condition

TL;DR

This paper demonstrates that iterative coding systems cannot exceed channel capacity unless their components are perfectly matched, extending the perfect matching condition to a broad class of channels and aiding capacity-achieving code design.

Contribution

It generalizes the perfect matching condition from binary erasure channels to all binary-input memoryless output-symmetric channels, providing a fundamental design criterion.

Findings

Iterative coding systems cannot surpass capacity without perfect component matching.

The perfect matching condition is extended to general binary-input memoryless channels.

Applications include constructing capacity-achieving codes and estimating required iterations.

Abstract

We show that iterative coding systems can not surpass capacity using only quantities which naturally appear in density evolution. Although the result in itself is trivial, the method which we apply shows that in order to achieve capacity the various components in an iterative coding system have to be perfectly matched. This generalizes the perfect matching condition which was previously known for the case of transmission over the binary erasure channel to the general class of binary-input memoryless output-symmetric channels. Potential applications of this perfect matching condition are the construction of capacity-achieving degree distributions and the determination of the number required iterations as a function of the multiplicative gap to capacity.