Grassmannians of codes
Ilaria Cardinali, Luca Giuzzi · Finite Fields and Their Applications · 2023
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∩Y is a [n,k−1]-linear code having minimum dual distance at least t+1. We are interested in the collinearity graph Λt(n,k) of Pt(n,k). The graph Λt(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δt(n,k) of the linear codes having minimum dual distance at least t+1 introduced in [9]. We shall study the structure of Λt(n,k) in relation to that of Δt(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ1(n,k) and Λ2(n,k) providing necessary and sufficient conditions for them to be connected.