The graphs of projective codes

TL;DR

This paper studies the structure of graphs formed by projective codes within Grassmann graphs, showing connectivity and diameter properties for large fields and exploring special cases like simplex codes.

Contribution

It establishes the connectivity and diameter equivalence of the projective code graphs to Grassmann graphs for large fields and provides insights into the structure of simplex code graphs.

Findings

The graph $mbda(n,k)_q$ is connected for $q \u2265 inom{n}{2}$.

The diameter of $mbda(n,k)_q$ matches that of the Grassmann graph.

Binary simplex codes of dimension 3 are maximal singular subspaces of a quadratic form.

Abstract

Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the field of $q$ elements ($1<k<n-1$) and denote by $\Pi(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$ codes. In the case when $q\ge \binom{n}{2}$, we show that the graph $\Pi(n,k)_q$ is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension $3$ are precisely maximal singular subspaces of a non-degenerate quadratic form.