On the Grassmann Graph of Linear Codes
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Let Γ(n,k) be the Grassmann graph formed by the k-dimensional subspaces of a vector space of dimension n over a field 𝔽 and, for t∈ℕ∖{0}, let Δ_t(n,k) be the subgraph of Γ(n,k) formed by the set of linear [n,k]-codes having minimum dual distance at least t+1. We show that if |𝔽|≥n t then Δ_t(n,k) is connected and it is isometrically embedded in Γ(n,k). This generalizes some results of [M. Kwiatkowski, M. Pankov, "On the distance between linear codes", Finite Fields Appl. 39 (2016), 251–263] and [M. Kwiatkowski, M. Pankov, A. Pasini, "The graphs of projective codes" Finite Fields Appl. 54 (2018), 15–29].
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