Finite-Connectivity Spin-Glass Phase Diagrams and Low Density Parity Check Codes

TL;DR

This paper explores the phase diagrams of finite connectivity spin-glasses and their relation to LDPC code performance, analyzing transitions with replica symmetry breaking to improve error correction methods.

Contribution

It extends the analysis of spin-glass phase diagrams to irregular systems and incorporates RSB theory to refine understanding of decoding thresholds for LDPC codes.

Findings

RS and RSB transition points differ significantly, affecting decoding thresholds.

RSB analysis predicts higher critical noise levels than RS, closer to Shannon limits.

Reentrant behavior of the dynamical transition line is altered under RSB.

Abstract

We obtain phase diagrams of regular and irregular finite connectivity spin-glasses. Contact is firstly established between properties of the phase diagram and the performances of low density parity check codes (LDPC) within the Replica Symmetric (RS) ansatz. We then study the location of the dynamical and critical transition of these systems within the one step Replica Symmetry Breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that, away from the Nishimori line, in the low temperature region, the location of the dynamical transition line does change within the RSB theory, in comparison with the (RS) case. For LDPC decoding over the binary erasure channel we find, at zero temperature and rate R=1/4 an RS critical transition point located at p_c = 0.67 while the critical RSB transition point is located at p_c = 0.7450, to be compared with the corresponding Shannon bound 1-R. For the binary symmetric channel (BSC) we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes within the RSB theory; the location of the dynamical transition point occurring at higher values of the channel noise. Possible practical implications to improve the performances of the state-of-the-art error correcting codes are discussed.