On null completely regular codes in Manhattan metric

TL;DR

This paper studies the existence and properties of completely regular codes in infinite grid graphs, identifying parameter families and employing linear programming to explore specific cases in 3 and 4 dimensions.

Contribution

It characterizes the existence of null completely regular codes in infinite grids and applies linear programming to analyze specific low-dimensional cases.

Findings

Certain parameter families of codes are derived from Hamming graphs or do not exist.

Binary linear programming can analyze codes in 3D and 4D grids for specific parameters.

The work advances understanding of code structures in infinite grid graphs.

Abstract

We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_\rho, where each set C_i, for 0<=i<=r-1, is an independent set. This work focuses on the existence problem for such codes in the n-dimensional infinite grid. We demonstrate that several parameter families of such codes necessarily arise from binary or ternary Hamming graphs or do not exist. Furthermore, employing binary linear programming techniques, we explore completely regular codes in infinite grids of dimensions 3 and 4 for the cases r=1 and r=2.