Codes on graphs: Models for elementary algebraic topology and statistical physics
For researchers in coding theory and statistical physics, this provides a unified framework connecting these fields, though it is primarily a tutorial with incremental new results.
This paper introduces graphical models of linear and group codes to bridge elementary algebraic topology and Ising-type statistical physics models, presenting new material on systematic group codes, homology/cohomology realizations, and dual/hybrid models.
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic (n,k) group codes and their information sets; normal realizations of homology and cohomology spaces; dual and hybrid models; and connections with system-theoretic concepts such as observability, controllability, and input/output realizations.