3 research outputs found

    On maximal cliques in the graph of simplex codes

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    The induced subgraph of the corresponding Grassmann graph formed by simplex codes is considered. We show that this graph, as the Grassmann graph, contains two types of maximal cliques. For any two cliques of the first type there is a monomial linear automorphism transferring one of them to the other. Cliques of the second type are more complicated and can contain different numbers of elements

    Grassmannians of codes

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    Consider the point line-geometry Pt(n,k){\mathcal P}_t(n,k) having as points all the [n,k][n,k]-linear codes having minimum dual distance at least t+1t+1 and where two points XX and YY are collinear whenever X∩YX\cap Y is a [n,k−1][n,k-1]-linear code having minimum dual distance at least t+1t+1. We are interested in the collinearity graph Λt(n,k)\Lambda_t(n,k) of Pt(n,k).{\mathcal P}_t(n,k). The graph Λt(n,k)\Lambda_t(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δt(n,k)\Delta_t(n,k) of the linear codes having minimum dual distance at least t+1t+1 introduced in~[M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263, doi:https://doi.org/10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of Λt(n,k)\Lambda_t(n,k) in relation to that of Δt(n,k)\Delta_t(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ1(n,k)\Lambda_1(n,k) and Λ2(n,k)\Lambda_2(n,k) providing necessary and sufficient conditions for them to be connected

    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 ïŹeld 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| ≄ nt then Δ t (n, k) is connected and it is isometrically embedded in Γ(n, k)
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