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Total perfect codes in Cayley sum graphs of cyclic groups

Masoumeh Koohestani, Doost Ali Mojdeh, Mohsen Ghasemi

TL;DR

The paper investigates total perfect codes in Cayley sum graphs CS$(G,S)$, focusing on cyclic groups and direct products of cyclic groups. It builds a bridge between total perfect codes and group factorizations (Vuza) and employs a multivariate polynomial framework to derive existence criteria. For cyclic groups, it establishes that when $|S|=k$ divides $n$ and the residues of $S$ are distinct modulo $k$, CS$(\mathbb{Z}_n,S)$ has a total perfect code, with explicit constructions such as $k\mathbb{Z}_n$, and analyzes periodic, aperiodic, and square-free cases via stabilizers and quotient reductions. The direct-product extension generalizes these results using a multivariate polynomial approach and explicit product-structure codes, while also noting that sufficiency may not be necessary, and detailing prime-power and square-free scenarios. These results advance the understanding of total perfect codes in algebraically structured networks and have potential implications for combinatorial design and coding theory in Cayley-type graph families.

Abstract

We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the connection set of the graph including when it is periodic, aperiodic, or square-free. To this end, we utilize a correspondence that we first establish between total perfect codes and factorizations of groups, along with their algebraic properties. We then generalize some of these conditions to the direct product of cyclic groups, i.e. $\mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_d}$.

Total perfect codes in Cayley sum graphs of cyclic groups

TL;DR

The paper investigates total perfect codes in Cayley sum graphs CS, focusing on cyclic groups and direct products of cyclic groups. It builds a bridge between total perfect codes and group factorizations (Vuza) and employs a multivariate polynomial framework to derive existence criteria. For cyclic groups, it establishes that when divides and the residues of are distinct modulo , CS has a total perfect code, with explicit constructions such as , and analyzes periodic, aperiodic, and square-free cases via stabilizers and quotient reductions. The direct-product extension generalizes these results using a multivariate polynomial approach and explicit product-structure codes, while also noting that sufficiency may not be necessary, and detailing prime-power and square-free scenarios. These results advance the understanding of total perfect codes in algebraically structured networks and have potential implications for combinatorial design and coding theory in Cayley-type graph families.

Abstract

We consider Cayley sum graphs over the cyclic group and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the connection set of the graph including when it is periodic, aperiodic, or square-free. To this end, we utilize a correspondence that we first establish between total perfect codes and factorizations of groups, along with their algebraic properties. We then generalize some of these conditions to the direct product of cyclic groups, i.e. .
Paper Structure (4 sections, 23 theorems, 21 equations, 2 figures, 1 table)

This paper contains 4 sections, 23 theorems, 21 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cayley Sum Graphs Over Cyclic Groups
  4. Cayley Sum Graphs over Direct Products of Cyclic Groups

Key Result

Lemma 2.1

Let $G$ be a group, $S$ a normal subset of $G$, and $C$ be a subgroup of $G$. Then, $C$ is a total perfect code of $\mathrm{CS}(G,S)$ of degree $|S|$ if and only if $G=C\oplus S$.

Figures (2)

  • Figure 1: $\mathrm{CS}(\mathbb{Z}_4\times \mathbb{Z}_4, \{ (0,1), (1,1), (1,3), (3,2)\})$
  • Figure 2: $\mathrm{CS}(\mathbb{Z}_3\times \mathbb{Z}_6, \{ (0,3), (0,1), (1,1)\})$

Theorems & Definitions (42)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 3.1
  • ...and 32 more