Generalized Hamming weights of projective Reed–Muller-type codes over graphs.

Let G be a connected graph and let X be the set of projective points defined by the column vectors of the incidence matrix of G over a field K of any characteristic. We determine the generalized Hamming weights of the Reed–Muller-type code over the set X in terms of graph theoretic invariants. As an application to coding theory we show that if G is non-bipartite and K is a finite field of char (K) ≠ 2 , then the r th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of G is the r th weak edge biparticity of G. If char (K) = 2 or G is bipartite, we prove that the r th generalized Hamming weight of that code is the r th edge connectivity of G. [ABSTRACT FROM AUTHOR]