On the Grassmann Graph of Linear Codes
Let Ξ(n,k) be the Grassmann graph formed by the k-dimensional subspaces of a vector space of dimension n over a field π½ and, for tβββ{0}, let Ξ_t(n,k) be the subgraph of Ξ(n,k) formed by the set of linear [n,k]-codes having minimum dual distance at least t+1. We show that if |π½|β₯n t then Ξ_t(n,k) is connected and it is isometrically embedded in Ξ(n,k). This generalizes some results of [M. Kwiatkowski, M. Pankov, "On the distance between linear codes", Finite Fields Appl. 39 (2016), 251β263] and [M. Kwiatkowski, M. Pankov, A. Pasini, "The graphs of projective codes" Finite Fields Appl. 54 (2018), 15β29].
READ FULL TEXT