Abstract
Decoding a redundant code, when redefined as a means for estimating a posterior probability distribution given a prior one, has a simple output if one looks for the posterior distribution in a form separable with respect to the information symbols. The optimal distribution of this form results from minimizing its cross-entropy with respect to the prior distribution, which for a linear code leads to a system of implicit analog equations to be solved. This can be done by iteration, but the system has multiple solutions so that determining proper initial values becomes the main difficulty. We interpret any linear code as a kind of product of its parity checks whose successive decodings provide adequate initial values for solving the system, provided the received symbols are reordered by decreasing reliability (except for few inversions in order to keep the generator matrix in systematic form). Simulation results are presented.
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G. BATTAIL, Le Décodage Pondéré en tant que Procédé de Réévaluation d'une Distribution de Probabilité, Colloque "3 journées sur le codage", Cachan, 24–26 nov. 1986; Annales Télécommunic., 42 no 9–10, sep.–oct. 1987, pp. 499–509
S. KULLBACK, Information Theory and Statistics, Wiley, 1959
J.E. SHORE and R.W. JOHNSON, Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy, IEEE Trans. Inf. Th., 26 no 1, jan. 1980, pp. 26–37
G. BATTAIL, J.C. BELFIORE and R. SFEZ, Decoding as a Means of Reestimating a Probability Distribution; Application to an Iterated Product of Parity-Check Codes, IEEE International Symposium on Information Theory, Kobe, Japan, 19–24 june 1988
G. BATTAIL and R. SFEZ, Suboptimum Decoding Using Kullback Principle, Conference IPMU, Urbino, Italy, 4–7 july 1988; Lecture Notes in Computer Science no 313, Springer, 1988, pp. 93–101
G. BATTAIL, Construction explicite de bons codes longs, submitted to the Annales Télécommunic.
P. ELIAS, Error-Free Coding, IRE Trans. on Inf. Th., jan. 1954, pp. 29–37
R.A. SILVERMAN and M. BALSER, Coding for Constant Data Rate Systems — Part I. A New Error-Correcting Code, Proc. IRE, sept. 1954, pp. 1428–1435
G. BATTAIL, Décodage Pondéré Optimal des Codes Linéaires en Blocs I.-Emploi Simplifié du Diagramme du Treillis, Annales Télécommunic., 38 no 11–12, nov.–dec. 1983, pp. 443–459
G. BATTAIL and M. DECOUVELAERE, Décodage par Répliques, Annales Télécommunic., 31 no 11–12, nov.–dec. 1976, pp. 387–404
G. BATTAIL and P. GODLEWSKI, Emploi de Représentations Polynomiales à Plusieurs Indéterminées pour le Décodage des Codes Redondants Linéaires, Annales Télécommunic., 33 no 3–4, mar.–apr. 1978, pp. 74–86
G. BATTAIL, M. DECOUVELAERE and P. GODLEWSKI, Replication Decoding, IEEE Trans. Inf. Th., 25 no 3, may 1979, pp. 332–345
G. BATTAIL and J. FANG, Décodage Pondéré Optimal des Codes Linéaires en Blocs II.-Analyse et Résultats de Simulation, Annales Télécommunic., 41 no 11–12, nov.–dec. 1986, pp. 580–604
J. FANG, G. COHEN, P. GODLEWSKI and G. BATTAIL, Le Décodage Pondéré des Codes Linéaires est NP-complet, Colloque "3 journées sur le codage", Cachan, 24–26 nov. 1986; Lecture Notes in Computer Science no 311, Springer, 1988, pp. 141–149
G. BATTAIL, Décodage par Résolution d'un Système d'Équations Implicites Analogiques, submitted to the Annales Télécommunic.
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© 1989 Springer-Verlag Berlin Heidelberg
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Battail, G. (1989). Weighted decoding of linear block codes by solving a system of implicit equations. In: Cohen, G., Wolfmann, J. (eds) Coding Theory and Applications. Coding Theory 1988. Lecture Notes in Computer Science, vol 388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0019852
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DOI: https://doi.org/10.1007/BFb0019852
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