Grassmannians of codes

04/17/2023

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by   I. Cardinali, et al.

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Consider the point line-geometry 𝒫_t(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 Λ_t(n,k) of 𝒫_t(n,k). The graph Λ_t(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δ_t(n,k) of the linear codes having minimum dual distance at least t+1 introduced in [M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251–263, doi:10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of Λ_t(n,k) in relation to that of Δ_t(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ_1(n,k) and Λ_2(n,k) providing necessary and sufficient conditions for them to be connected.