On the Grassmann graph of linear codes

Abstract

Let Γ(n, k) be the Grassmann graph formed by the kdimensional subspaces of a vector space of dimension n over a field 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| ≥ n t then Δ t (n, k) is connected and it is isometrically embedded in Γ(n, k).

On the graph of non-degenerate linear $[n,2]_2$ codes

“…, g 1 (P n−1 ) are mutually distinct. Then S c (P i ) ∩ S c (P j ) = {P i + P j }, S c (g 1 (P i )) ∩ S c (g 1 (P j )) = {f (P i + P j )} which means that (3) f (P i + P j ) = g 1 (P i ) + g 1 (P j ), i, j ∈ {1, . .…”

“…In contrast to the (n, k, q)-Grassmann graph, the structure of this graph essentially depends on the parameters n, k, q (structural properties holding for some triples n, k, q fail for others). Induced subgraphs of the (n, k, q)-Grassmann graph corresponding to different types of linear codes (projective and simplex codes, codes with lower bounded minimal dual distance) are considered in [3,8,9]. By Chow's theorem [4], every automorphism of a Grassmann graph (over an arbitrary field) is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space (the second possibility is realized only in the case when the dimension of subspaces is the half of the dimension of the vector space).…”

The graphs of non-degenerate linear codes

“…By [7], the distances in this graph coincide with the distances in the Grassmann graph if and only if n < (q + 1) 2 + k − 2; maximal cliques and automorphisms are determined in [8]. Subgraphs of the Grassmann graph formed by different families of linear codes are considered in [3,9,10]. By Chow's theorem, every automorphism of the Grassmann graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space and the second possibility is realized only for n = 2k.…”

Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

“…Graphs and geometries related to various types of linear codes are considered in [3,4,6,7,8,9,11]. In the present paper, we investigate point-line geometries whose maximal singular subspaces correspond to equivalence classes of binary equidistant codes (not necessarily non-degenerate).…”