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Completely regular codes with covering radius 1 and the second eigenvalue in 3-dimensional Hamming graphs

Ivan Mogilnykh, Anna Taranenko, Konstantin Vorob'ev

Abstract

We obtain a classification of the completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(3,q) up to q and intersection array. Due to works of Meyerowitz, Mogilnykh and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(n,q) for any n and completely regular codes with covering radius 1 in H(3,q).

Completely regular codes with covering radius 1 and the second eigenvalue in 3-dimensional Hamming graphs

Abstract

We obtain a classification of the completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(3,q) up to q and intersection array. Due to works of Meyerowitz, Mogilnykh and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(n,q) for any n and completely regular codes with covering radius 1 in H(3,q).
Paper Structure (7 sections, 22 theorems, 24 equations, 2 tables)

This paper contains 7 sections, 22 theorems, 24 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Stochastic properties of CRCs with $\rho=1$ in the Cartesian products of complete graphs
  4. Constructions of completely regular codes in $3$-dimensional Hamming graphs
  5. CRCs in $H(3,q)$ with clique property
  6. CRCs with the second eigenvalue and small odd $\gamma$
  7. The parameters of CRCs with covering radius $1$ and the second eigenvalue in $H(n,q)$

Key Result

Proposition 1

Let $C$ be a code in $H(n,q)$ with nonessential position $j$. The code $C$ is completely regular with covering radius $\rho$, intersection array $A$ and eigenvalues $\{\lambda_i(n,q):i\in I\}$ if and only if the code in $H(n-1,q)$ obtained by deleting $j$th position in the tuples of $C$ is completel

Theorems & Definitions (41)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 31 more