Some results on perfect codes in Cayley sum graphs

TL;DR

The paper addresses the problem of characterizing subgroup perfect codes in Cayley sum graphs and their relation to quotient groups. It proves a necessary and sufficient condition: a normal subgroup $H$ is a subgroup perfect code of $G$ in a Cayley sum graph iff $G$ has a normal subset $X$ with $X \cup \{1\}$ as a left transversal of $H$; it also derives a nonexistence criterion via cosets containing no central elements. Applying the framework to the nonabelian group $V_{8n}$ (with odd $n$), the authors classify nontrivial subgroup perfect codes as $H \in \{\\langle a^n\\rangle, \\langle a^n b^2\\rangle, \\langle a^j b\\rangle, \\langle a^j b^3\\rangle\\}$ with $j$ odd, totaling $2n+2$ such subgroups, and provide explicit graphs realizing these codes alongside a parallel classification for total perfect codes. An accompanying computational appendix supports the constructions by detailing conjugacy classes, left cosets, and transversals in $V_{8n}$.

Abstract

We establish a necessary and sufficient condition for a normal subgroup of a finite group to be a subgroup perfect code.

Figures (2)

• Figure 1: $\mathrm{CS}(D_{12},\{a,a^5,ab,a^3b,a^5b\})$
• Figure 2: $\mathrm{CS}(\langle a \rangle ,\{a,a^5\})$

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• proof
• Example 3
• Theorem 4
• proof
• Remark 5
• Theorem 6
• proof
• Remark 7