In this talk, we give an overview of Seymour's matroid decomposition theory 
in the context of binary linear codes, and discuss some of its implications 
for linear programming (LP) decoding of a binary linear code. Given 
the formulation by Feldman \emph{et al.} of maximum-likelihood (ML) decoding
over a discrete memoryless channel as an LP problem, we 
translate matroid-theoretic results of Gr\"otschel and Truemper 
from the combinatorial optimization literature as examples of 
non-trivial families of codes for which ML decoding can be implemented in time 
polynomial in the length of the code. One such family is that consisting 
of codes $C$ for which the codeword polytope is identical to the 
Koetter-Vontobel fundamental polytope derived from the entire dual code 
$C^\perp$. However, we also show that such families of codes are not good 
in a coding-theoretic sense --- either their dimension or their 
minimum distance must grow sub-linearly with codelength.