Weight vs. Magnetization Enumerator for Gallager Codes

We propose a method to determine the critical noise level for decoding Gallager type low density parity check error correcting codes. The method is based on the magnetization enumerator (M), rather than on the weight enumerator (W) presented recently in the information theory literature. The interpretation ofour method is appealingly simple, and the relation between the different decoding schemes such as typical pairs decoding, MAP, and finite temperature decoding (MPM) becomes clear. Our results are more optimistic than those derived via the methods ofinf ormation theory and are in excellent agreement with recent results from another statistical physics approach.

[1]  Robert J. McEliece,et al.  BSC Thresholds for Code Ensembles Based on “Typical Pairs” Decoding , 2001 .

[2]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[3]  D. Saad,et al.  Error-correcting code on a cactus: A solvable model , 2000, cond-mat/0005109.

[4]  D. Saad,et al.  Tighter decoding reliability bound for Gallager's error-correcting code. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  David Mackay,et al.  On Thresholds of Codes , 2000 .

[6]  H. Nishimori,et al.  Statistical mechanics of image restoration and error-correcting codes. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  H. Nishimori.  Statistical Physics of Spin Glasses and Information Processing , 2001 .

[8]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[9]  T. Richardson,et al.  Design of provably good low-density parity check codes , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[10]  Saad,et al.  Typical performance of gallager-type error-correcting codes , 2000, Physical review letters.

[11]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[12]  Yoshiyuki Kabashima,et al.  Statistical mechanics of typical set decoding. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[14]  Y. Iba.  The Nishimori line and Bayesian statistics , 1998, cond-mat/9809190.

[15]  M. Opper,et al.  Advanced mean field methods: theory and practice , 2001 .