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On null completely regular codes in Manhattan metric

I. Yu. Mogilnykh, A. Yu. Vasil'eva

TL;DR

This work analyzes null completely regular codes (CRC) in the $n$-dimensional infinite grid $G_n$, focusing on $r$-null CRCs with $a_0=\cdots=a_{r-1}=0$ and related all-zeros/all-ones variants. It combines constructive liftings from binary/ternary Hamming graphs and related Cayley graphs with a binary linear programming framework to characterize existence and nonexistence of CRCs in $G_n$, especially for $n=3,4$ and radii $\rho\le 2$. The authors derive extensive classifications for $1$-null, $2$-null, and $3$-null CRCs, establish period-4 behavior in several constructions, and identify precise parameter matrices that yield CRCs versus those ruled out by the LP-based methods. The study links CRC existence in the grid to classical codes (Golay, BCH, Preparata) via liftings to Hamming graphs, and provides a practical LP toolkit to certify nonexistence or existence of CRCs in small-dimension grids, advancing understanding of CRC structures on infinite graphs with Manhattan metric.

Abstract

We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_ρ, 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.

On null completely regular codes in Manhattan metric

TL;DR

This work analyzes null completely regular codes (CRC) in the -dimensional infinite grid , focusing on -null CRCs with and related all-zeros/all-ones variants. It combines constructive liftings from binary/ternary Hamming graphs and related Cayley graphs with a binary linear programming framework to characterize existence and nonexistence of CRCs in , especially for and radii . The authors derive extensive classifications for -null, -null, and -null CRCs, establish period-4 behavior in several constructions, and identify precise parameter matrices that yield CRCs versus those ruled out by the LP-based methods. The study links CRC existence in the grid to classical codes (Golay, BCH, Preparata) via liftings to Hamming graphs, and provides a practical LP toolkit to certify nonexistence or existence of CRCs in small-dimension grids, advancing understanding of CRC structures on infinite graphs with Manhattan metric.

Abstract

We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_ρ, 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.
Paper Structure (10 sections, 18 theorems, 29 equations, 1 table)

This paper contains 10 sections, 18 theorems, 29 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Constructions of CRCs in $G_n$
  4. General constructions
  5. Null-CRC codes
  6. Local structure of $n$-dimensional infinite grid
  7. Null CRCs with $c_2\leq 3$
  8. $1$-null CRCs with $a_{1}=1$
  9. All-ones CRC
  10. Binary linear programming for null CRC in $G_3$ and $G_4$

Key Result

Theorem 1

AvgVas22 The following holds for a CRC in $G_n$ with parameter matrix $[a_0,b_0 | c_1,a_1,b_1 | c_2,a_2,b_2 | \ldots | c_{\rho},a_{\rho}]$ and covering radius $\rho$:

Theorems & Definitions (34)

  • Theorem 1
  • Proposition 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • ...and 24 more