Code Decomposition: Theory and Applications

In this paper, 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. As shown by Feldman et al. maximum-likelihood (ML) decoding over a discrete memoryless channel can be formulated as an LP problem. Using this formulation, we translate matroid-theoretic results of Grotschel 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. 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.