The glassy phase of Gallager codes

Details The glassy phase of Gallager codes The glassy phase of Gallager codes

2001-04-04

arxiv.org/abs/cond-mat/0104079v1

Eur. Phys. J. B 23, 121 (2001)

Physics

Condensed Matter

Disordered Systems and Neural Networks

27 pages, 8 eps figures

Scientific paper

10.1007/s100510170089

Gallager codes are the best error-correcting codes to-date. In this paper we study them by using the tools of statistical mechanics. The corresponding statistical mechanics model is a spin model on a sparse random graph. The model can be solved by elementary methods (i.e. without replicas) in a large connectivity limit. For low enough temperatures it presents a completely frozen glassy phase (q_{EA}=1). The same scenario is shown to hold for finite connectivities. In this case we adopt the replica approach and exhibit a one-step replica symmetry breaking order parameter. We argue that our ansatz yields the exact solution of the model. This allows us to determine the whole phase diagram and to understand the performances of Gallager codes.

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