Deterministic Constructions for Large Girth Protograph LDPC Codes

TL;DR

This paper analyzes and constructs protograph LDPC codes with large girth that achieve exponential decay in bit-error probability below the threshold, improving block-error thresholds and enabling applications like secure communication.

Contribution

It provides deterministic constructions of large girth protograph LDPC codes with improved block-error thresholds and analyzes their density evolution over the binary erasure channel.

Findings

Bit-error probability decreases double exponentially below threshold.

Deterministic constructions achieve girth logarithmic in blocklength.

Optimized protographs outperform standard ensemble in block-error threshold.

Abstract

The bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity, if there is a non-zero fraction of degree-two bit nodes. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.