Codes on graphs: Models for elementary algebraic topology and statistical physics

TL;DR

This paper introduces elementary algebraic topology concepts and their applications to statistical physics models using graphical codes, with new insights into group codes, homology, and system theory connections.

Contribution

It provides new systematic methods for (n,k) group codes, realizations of homology spaces, and links between algebraic topology and system theory.

Findings

New systematic (n,k) group codes and information sets

Normal realizations of homology and cohomology spaces

Connections between algebraic topology and system-theoretic concepts

Abstract

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.