Large deviations for coding Markov chains and Gibbs random fields

Fixed block coding schemes for Gibbs random fields are proposed in which the empirical expectations of the local interactions of the Gibbs measure is compared to its expectation with respect to all Gibbs measures having those interactions. The exponential decay of the error probabilities is proven and it is shown that the code rates equal the entropy of the random field. In addition, it is shown that any coding scheme based on regarding the field as a 1-D sequence of symbols has rate greater than the entropy of the field. The theory of fixed length coding is approached from the point of view of large deviations, both for the calculation of the error exponents or error probabilities and for the calculation of the encoding rates or the asymptotic combinatorics of the coding schemes. This approach is also applied to fixed length coding schemes of Markov sources for which estimates on the error exponents and on the rates are derived