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