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.
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