Codes on Graphs: Models for Elementary Algebraic Topology and Statistical Physics

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.