The Distribution of Cycle Lengths in Graphical Models for Iterative Decoding

TL;DR

This paper investigates the distribution of cycle lengths in turbo and LDPC decoding graphs, providing probabilistic estimates and simulations that reveal how cycle lengths impact iterative decoding performance.

Contribution

It introduces a probabilistic model for cycle length distribution in turbo and LDPC graphs, validated by simulations, and analyzes the impact of permutation strategies on cycle structures.

Findings

Probability of short cycles decreases with block length

S-random permutation eliminates very short cycles but not longer ones

Cycle length distribution insights help explain decoding success

Abstract

This paper analyzes the distribution of cycle lengths in turbo decoding and low-density parity check (LDPC) graphs. The properties of such cycles are of significant interest in the context of iterative decoding algorithms which are based on belief propagation or message passing. We estimate the probability that there exist no simple cycles of length less than or equal to k at a randomly chosen node in a turbo decoding graph using a combination of counting arguments and independence assumptions. For large block lengths n, this probability is approximately e^{-{2^{k-1}-4}/n}, k>=4. Simulation results validate the accuracy of the various approximations. For example, for turbo codes with a block length of 64000, a randomly chosen node has a less than 1% chance of being on a cycle of length less than or equal to 10, but has a greater than 99.9% chance of being on a cycle of length less than or equal to 20. The effect of the "S-random" permutation is also analyzed and it is shown that while it eliminates short cycles of length k<8, it does not significantly affect the overall distribution of cycle lengths. Similar analyses and simulations are also presented for graphs for LDPC codes. The paper concludes by commenting briefly on how these results may provide insight into the practical success of iterative decoding methods.