Graphs related to $2$-dimensional simplex codes

TL;DR

This paper characterizes the distance relations and symmetries in the graph of 4-ary 2-dimensional simplex codes, revealing its structure and diameter.

Contribution

It provides a complete description of the distance relations and symmetries in the graph of 4-ary 2-dimensional simplex codes, a previously unexplored structure.

Findings

The graph is connected with diameter 3.

Distances from each vertex are fully characterized.

Symmetries of the distance sets are described.

Abstract

We give a complete description of the distance relation on the graph of $4$-ary simplex codes of dimension $2$. This is a connected graph of diameter $3$. For every vertex we determine the sets of all vertices at distance $i\in\{1,2,3\}$ and describe their symmetries.