3 research outputs found
On maximal cliques in the graph of simplex codes
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
Consider the point line-geometry having as points all the -linear codes having minimum dual distance at least and where two points and are collinear whenever is a -linear code having minimum dual distance at least . We are interested in the collinearity graph of The graph is a subgraph of the Grassmann graph and also a subgraph of the graph of the linear codes having minimum dual distance at least 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 in relation to that of and we will characterize the set of its isolated vertices. We will then focus on and providing necessary and sufficient conditions for them to be connected
On the Grassmann graph of linear codes
Let Î(n, k) be the Grassmann graph formed by the k-
dimensional subspaces of a vector space of dimension n over a
ïŹeld F and, for t â N {0}, let Î t (n, k) be the subgraph
of Î(n, k) formed by the set of linear [n, k]-codes having
minimum dual distance at least t +1. We show that if |F| â„ nt
then Î t (n, k) is connected and it is isometrically embedded
in Î(n, k)