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

摘要

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) not equal 2, then the rth generalized Hamming weight of the linear code generated by the rows of the incidence matrix of G is the rth weak edge biparticity of G. If char(K) = 2 or G is bipartite, we prove that the rth generalized Hamming weight of that code is the rth edge connectivity of G. (C) 2019 Elsevier B.V. All rights reserved.