On the distance between linear codes

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements and let ${\Gamma }_k(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of $V$, $1<k<n-1$. Denote by $\Gamma (n,k{)}_q$ the restriction of ${\Gamma }_k(V)$ to the set of all non-degenerate linear $[n,k{]}_q$ codes. We show that for any two codes the distance in $\Gamma (n,k{)}_q$ coincides with the distance in ${\Gamma }_k(V)$ only in the case when $n<(q+1{)}^2+k-2$, i.e. if $n$ is sufficiently large then for some pairs of codes the distances in the graphs ${\Gamma }_k(V)$ and $\Gamma (n,k{)}_q$ are distinct. We describe one class of such pairs.