Establishment of statistical mechanical methods in information sciences

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We have developed a framework to evaluate performance of various coding methods such as error correcting and compression codes, and CDMA multiuser communication systems utilizing notions and techniques of statistical mechanics. The developed framework is composed mainly of the following three schemes :1. Scheme for assessing typical performance of codes infinite length Regarding a given code as a kind of spin glass models, this scheme makes it possible to characterize the typical performance of the code as the length of the codes tends to infinity utilizing the concept of phase transition of disordered systems.2. Scheme for characterizing atypical behavior of codes of finite length Combining evaluation methods of large deviation analysis and the replica method that has been developed in statistical mechanics, this scheme enables us to assess atypical behavior of an objective code utilizing a certain exponent.3. Scheme for evaluating typical behavior of belief propagation Belief propagation is an inference algorithm that currently occupies a special status in research of codes as a standard decoding algmithm of modem error correcting codes. We have developed a scheme that makes it possible to characterize typical behavior of the algorithm utilizing macroscopic quantities, in particula4 for systems of dense connectivity. In addition, we have also obtained several results concerning nature of statistical mechanical methods :4. We have discovered a model system of finite system size for which analytical continuation of the replica number can be exactly performed, which provides complete information about the origin of the replica symmetry breaking of a certain type.5. We have gained a plausible conjecture that the condition of the de Almeida-Thouless instability known in static analysis of spin glass systems generally accords with the dynamical instability condition of the fixed point of the belief propagation algorithm in the case of mean field models..