## DIMACS TR: 99-54

## On Polynomial Invariants of Codes, Matroids and Finite Interaction Models

### Author: Alexander Barg

**
ABSTRACT
**

A linear code can be thought of as a vector matroid
represented by the columns of code's generator matrix;
a well-known result in this context is Greene's
theorem on a connection of the weight polynomial of the code
and the Tutte polynomial of the matroid. We examine this
connection from the coding-theoretic viewpoint, building
upon the rank polynomial of the code.
This enables us

- to relate the weight polynomial of codes
and the reliability polynomial of linear matroids and to prove bounds
on the latter;
- to prove that the partition polynomial of the Potts model equals
the weight polynomial of the cocycle code of the underlying graph,
and
- to give a simple proof of Greene's theorem and its generalization.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-54.ps.gz

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