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.