Optimal encoding on discrete lattice with translational invariant constrains using statistical algorithms

Abstract: In this paper will be presented methodology of encoding information in valuations of discrete lattice with some translational invariant constrains in asymptotically optimal way. The method is based on finding statistical description of such valuations and changing it into statistical algorithm, which allows to construct deterministically valuation with given statistics. Optimal statistics allow to generate valuations with uniform distribution - we get maximum information capacity this way. It will be shown that we can reach the optimum for one-dimensional models using maximal entropy random walk and that for the general case we can practically get as close to the capacity of the model as we want (found numerically: lost 10^{-10} bit/node for Hard Square). There will be also presented simpler alternative to arithmetic coding method which can be used as cryptosystem and data correction method too.