Eur. Phys. J. B 23, 121-136 (2001)
https://doi.org/10.1007/s100510170089

The glassy phase of Gallager codes

Laboratoire de Physique Théorique de l'École Normale Supérieure (UMR 8549 CNRS-École Normale Supérieure) , 24 rue Lhomond, 75231 Paris Cedex 05, France

Corresponding author: ^a andrea.montanari@lpt.ens.fr

Received: 9 April 2001
Published online: 15 September 2001

Abstract

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.