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

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.