On the Convex Geometry of Binary Linear Codes

A code polytope is defined to be the convex hull in R n of the points in {0, 1}n corresponding to the codewords of a binary linear code. This paper contains a collection of results concerning the structure of such code polytopes. A survey of known results on the dimension and the minimal polyhedral representation of a code polytope is first presented. We show how these results can be extended to obtain the complete facial structure of the polytope determined by the [n, n−1] even-weight code. We then give a result classifying the types of 3-faces a general code polytope can have, which shows that the faces of such a polytope cannot be completely arbitrary. Finally, we show how geometrical arguments lead to a simple lower bound on the number of minimal codewords of a code, and characterize the codes for which this bound is attained with equality. This also yields an interesting intermediate result that classifies simple code polytopes. The motivation for our study of code polytopes comes from the formulation by Feldman, Wainwright and Karger of maximum-likelihood decoding as a linear programming problem over the code polytope.