Connectedness of projective codes in the Grassmann graph

TL;DR

This paper proves that the set of projective linear codes forms a connected subgraph within the Grassmann graph, using projective systems and elementary linear algebra, highlighting structural properties of these codes.

Contribution

It establishes the connectedness of projective codes in the Grassmann graph, a new insight into the geometric structure of linear codes.

Findings

Projective $[n,k]_q$ codes form a connected subgraph in the Grassmann graph.

Utilizes projective systems and elementary linear algebra for the proof.

Provides a geometric perspective on the structure of linear codes.

Abstract

Using the concept of projective systems for linear codes and elementary linear algebra, we show that projective $[n,k]_q$ codes form a connected subgraph in the Grassmann graph consisting of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field.