Grassmannians of codes

Consider the point line-geometry Pt(n, k) having as points all the [n, k]-linear codes having minimum dual distance at least t + 1 and where two points X and Y are collinear whenever X boolean AND Y is a [n, k -1]-linear code having minimum dual distance at least t + 1. We are interested in the collinearity graph Lambda t(n, k) of Pt(n, k). The graph Lambda t(n, k) is a subgraph of the Grassmann graph and also a subgraph of the graph Delta t(n, k) of the linear codes having minimum dual distance at least t +1 introduced in [9]. We shall study the structure of Lambda t(n, k) in relation to that of Delta t(n, k) and we will characterize the set of its isolated vertices. We will then focus on Lambda 1(n, k) and Lambda 2(n, k) providing necessary and sufficient conditions for them to be connected. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://