On the distance between linear codes

TL;DR

This paper investigates the relationship between distances in the Grassmann graph of all subspaces and the graph restricted to non-degenerate linear codes, revealing conditions where these distances differ.

Contribution

It establishes the precise conditions under which the distances in the two graphs coincide or differ, and describes a class of code pairs where the distances are distinct.

Findings

Distances coincide when n < (q+1)^2 + k - 2

For larger n, some code pairs have different distances in the two graphs

Identifies a specific class of code pairs with differing distances

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.